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PROFESSOR: Hi.
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Our lecture today actually has
us backtrack a little bit to a
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concept which is actually more
fundamental than that of
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differentiability.
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It's a topic which is called
'continuous functions'.
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In other words, our lecture
today is concerned with a
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topic called Continuity.
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00:00:52,440 --> 00:00:56,730
And actually, the topic of
continuity had its roots way
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00:00:56,730 --> 00:01:00,150
back at the beginning of our
course when we first raised
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00:01:00,150 --> 00:01:03,970
the question: does the limit of
'f of x' as 'x' approaches
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'a' equal 'f of a'?
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00:01:05,950 --> 00:01:07,990
Remember, when we first started
talking about the
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limit concept, our intuitive
approach was to say, look-it,
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as 'x' gets arbitrarily close
to 'a', 'f of x' gets
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00:01:14,440 --> 00:01:16,550
arbitrarily close to 'f of a'.
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00:01:16,550 --> 00:01:19,610
And we saw that this particular
definition had a
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00:01:19,610 --> 00:01:22,850
few loopholes in it, even though
intuitively this is
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00:01:22,850 --> 00:01:24,380
what we would have liked.
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00:01:24,380 --> 00:01:28,110
And so what I want to do today
is to see what happens if we
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can be sure that the limit of 'f
of x' as 'x' approaches 'a'
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00:01:31,580 --> 00:01:32,420
equals 'f of a'.
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00:01:32,420 --> 00:01:34,500
What is implied if this happens
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00:01:34,500 --> 00:01:36,490
to be a true statement?
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00:01:36,490 --> 00:01:41,930
And my first claim is that it is
implicitly implied that 'f
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00:01:41,930 --> 00:01:43,980
of a' must at least
make sense.
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00:01:43,980 --> 00:01:46,720
Otherwise, we wouldn't have
written it over here.
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00:01:46,720 --> 00:01:51,650
In other words, 'f' must be
defined when 'x' equals 'a'.
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00:01:51,650 --> 00:01:53,910
'f of a' must be defined.
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00:01:53,910 --> 00:01:57,050
Now to correlate that with
material that we've had
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00:01:57,050 --> 00:01:59,660
previously, let's go back
to an example that we've
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00:01:59,660 --> 00:02:02,570
discussed before, or at least
an example which is close to
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00:02:02,570 --> 00:02:04,830
something we've studied
before.
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00:02:04,830 --> 00:02:10,320
Let's define 'f of x' to be 'x
squared minus 1' over 'x - 1'.
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00:02:10,320 --> 00:02:13,410
Now you see as long as 'x' is
not equal to 1, we can cancel
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'x - 1' from both numerator and
denominator, leaving this
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00:02:17,260 --> 00:02:19,040
equal to 'x + 1'.
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00:02:19,040 --> 00:02:20,050
And we then see what?
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00:02:20,050 --> 00:02:22,390
That the limit of 'f of
x' as 'x' approaches
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00:02:22,390 --> 00:02:25,270
1 is equal to 2.
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00:02:25,270 --> 00:02:30,930
On the other hand, notice that
'f of 1' turns out to be 0/0
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00:02:30,930 --> 00:02:33,180
in this case, which
is undefined.
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00:02:33,180 --> 00:02:36,260
In other words, in this
particular case, the limit of
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00:02:36,260 --> 00:02:40,930
'f of x' as 'x' approaches 1
is not equal to 'f of 1' if
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only because 'f of 1'
isn't even defined.
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00:02:44,340 --> 00:02:47,480
And again, by way of a review,
let's look at this thing
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00:02:47,480 --> 00:02:48,790
pictorially.
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00:02:48,790 --> 00:02:54,800
Remember, 'x squared minus 1'
over 'x - 1' is 'x + 1' except
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when 'x' equals 1.
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00:02:56,800 --> 00:03:00,310
Consequently, to graph 'y'
equals 'x squared minus 1'
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00:03:00,310 --> 00:03:05,750
over 'x - 1', we simply graph
'y' equals 'x + 1' with the
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point corresponding to 'x'
equals 1, meaning the point 1
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comma 2 deleted.
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In other words, this point
deleted is the graph 'y'
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equals 'x squared minus
1' over 'x - 1'.
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You see again there is no
definition at 'x' equals 1
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because this point is missing.
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It would be the 0/0 form.
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Now, to be sure, we can clean
this up in the sense that we
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could define a new function 'g
of x' to be 'x squared minus
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00:03:38,130 --> 00:03:44,250
1' over 'x - 1' as long as 'x'
is not equal to 1 and define
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00:03:44,250 --> 00:03:47,620
it to be 2 when 'x' equals
1, which, by the way,
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geometrically is just a fancy
way of writing the equation of
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the line 'y' equals 'x + 1'.
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In other words, all we're saying
is that letting 'g of
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x' be 2 when 'x' equals 1 plugs
the little hole to the
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periods over here.
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00:04:02,960 --> 00:04:06,270
Now again, I hope that this
seems fairly familiar from our
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discussion on limits, but for
the time being, all I want us
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to see is that as soon as you
write down that the limit of
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'f of x' as 'x' approaches 'a'
equals 'f of a', you at least
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00:04:16,700 --> 00:04:18,920
imply that 'f of a'
must be defined.
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00:04:18,920 --> 00:04:22,019
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00:04:22,019 --> 00:04:24,850
Let's see another property
that's implied by our
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00:04:24,850 --> 00:04:26,370
definition.
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I claim, as you might expect,
that if the limit of 'f of x'
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00:04:31,820 --> 00:04:35,170
as 'x' approaches 'a' is 'f of
a', that means that 'f of x'
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00:04:35,170 --> 00:04:38,290
is near 'f of a' when
'x' is near 'a'.
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In other words, my claim is that
the curve 'y' equals 'f
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of x' is unbroken in
a neighborhood
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00:04:43,820 --> 00:04:45,350
of 'x' equals 'a'.
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00:04:45,350 --> 00:04:48,280
And to see what I mean by
that, let's look at this
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00:04:48,280 --> 00:04:49,340
picture over here.
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Suppose, for example, that our
curve 'y' equals 'f of x'--
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let me darken this up so that we
can see it a little better.
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00:04:56,090 --> 00:05:01,340
Suppose that curve had a break
at 'x' equals 'a', OK?
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Now, let's say 'f of
a' is over here.
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What I could do is now pick an
interval surrounding 'f of a',
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00:05:08,440 --> 00:05:13,190
if there was a break over here,
such that this band
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never touched or included
the top curve here.
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00:05:17,090 --> 00:05:20,630
Now notice, from this point of
view, that no matter how close
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00:05:20,630 --> 00:05:24,860
'x' gets to 'a', as long as 'x'
is greater than 'a', 'f of
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00:05:24,860 --> 00:05:28,240
x', which is up here on this
dark curve, can never be
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within epsilon of 'f of a'.
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In other words, if there's a
break in the curve I could
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00:05:35,610 --> 00:05:38,340
always fix it up so that the
limit of 'f of x' as 'x'
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approaches 'a' is not
equal to 'f of a'.
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At any rate, with these two
properties as motivation, let
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us now give our basic
definition.
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And the basic definition is
simply this: a function 'f' is
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called continuous that 'x'
equals 'a' precisely if the
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00:05:57,810 --> 00:06:00,800
property that we were discussing
is present, namely,
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00:06:00,800 --> 00:06:04,170
if the limit of 'f of x' as
'x' approaches 'a' is
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00:06:04,170 --> 00:06:07,050
equal to 'f of a'.
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00:06:07,050 --> 00:06:09,910
We generalize this definition
and say 'f' is called
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00:06:09,910 --> 00:06:15,640
continuous on the entire
interval 'I', or on the
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00:06:15,640 --> 00:06:19,530
interval 'I', if the limit of 'f
of x' as 'x' approaches 'a'
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is equal to 'f of a' for
each 'a' in 'I'.
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In other words, notice that
our first definition is a
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00:06:28,030 --> 00:06:29,030
local property.
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Namely, we define continuous or
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00:06:31,360 --> 00:06:35,090
continuity at a given value.
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00:06:35,090 --> 00:06:37,940
And then if the function happens
to be continuous at
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every point, then we call the
function itself continuous.
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00:06:42,110 --> 00:06:45,700
What the thing means pictorially
is that if the
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function is continuous at 'x'
equals 'a', it means that in
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00:06:49,300 --> 00:06:52,100
terms of the graph of the
function, that in a
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neighborhood of the point 'a'
comma 'f' of a on the graph,
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in a neighborhood of that point,
in other words, in a
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00:06:58,720 --> 00:07:02,230
sufficiently small interval
surrounding that point, the
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00:07:02,230 --> 00:07:06,870
curve itself must be unbroken.
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00:07:06,870 --> 00:07:09,510
Now, you see, this may
sound trivial.
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00:07:09,510 --> 00:07:12,220
Remember, when we first started
talking about limits,
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we had the feeling that limit of
'f of x' as 'x' approaches
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00:07:14,960 --> 00:07:17,470
a should always equal
'f of a'.
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00:07:17,470 --> 00:07:21,130
And the reason for this was that
instinctively we seem to
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00:07:21,130 --> 00:07:23,260
always think of continuous
functions.
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00:07:23,260 --> 00:07:24,840
We think of what?
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00:07:24,840 --> 00:07:29,070
Things changing in such a way
the graph of the change is
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00:07:29,070 --> 00:07:32,280
unbroken even though, of course,
there are places where
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00:07:32,280 --> 00:07:34,660
discontinuities occur.
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00:07:34,660 --> 00:07:37,335
At any rate, what I'm saying
is that many of the results
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00:07:37,335 --> 00:07:39,920
that I now want to discuss
with you, which are very
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00:07:39,920 --> 00:07:42,940
important properties of
continuous functions, may seem
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00:07:42,940 --> 00:07:46,500
self-evident because we keep
thinking about any function as
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00:07:46,500 --> 00:07:47,600
being continuous.
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00:07:47,600 --> 00:07:50,410
But I will try to emphasize the
fact that these properties
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00:07:50,410 --> 00:07:53,670
are not true if the function
is not continuous.
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00:07:53,670 --> 00:07:58,260
I also would like to show a
balance between geometric
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00:07:58,260 --> 00:08:00,500
ideas and analytic ideas.
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In other words, let me start off
by seeing what things seem
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00:08:03,630 --> 00:08:07,770
to follow about continuous
functions based primarily on
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00:08:07,770 --> 00:08:09,550
the graph idea.
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00:08:09,550 --> 00:08:13,340
For example, I claimed that
continuous functions assume
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00:08:13,340 --> 00:08:16,990
their maximum and minimum values
on any closed interval.
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00:08:16,990 --> 00:08:20,880
Now, this sounds like a big
mouthful, and it also sounds
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00:08:20,880 --> 00:08:21,650
kind of trivial.
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00:08:21,650 --> 00:08:24,350
You say doesn't any function
have a maximum and minimum
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00:08:24,350 --> 00:08:26,440
value on a closed interval?
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00:08:26,440 --> 00:08:27,360
And the answer is no.
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00:08:27,360 --> 00:08:30,500
For example, take the
curve 'y' equals '1
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00:08:30,500 --> 00:08:32,630
over ''x - 1' squared''.
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00:08:32,630 --> 00:08:36,700
Notice that when 'x' is 1, 'y'
is infinite, which means that
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00:08:36,700 --> 00:08:40,039
if we graph this particular
function, we find that the
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00:08:40,039 --> 00:08:42,090
graph does something
like this.
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00:08:42,090 --> 00:08:43,770
In other words, it goes--
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00:08:43,770 --> 00:08:47,100
it jumps up here, comes
back, et cetera.
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00:08:47,100 --> 00:08:52,400
Now, what is the highest value
or what is the highest point
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00:08:52,400 --> 00:08:53,510
on our graph here?
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00:08:53,510 --> 00:08:56,070
Well, notice that the graph
is broken here.
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00:08:56,070 --> 00:08:58,390
It's discontinuous when
'x' equals 1.
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00:08:58,390 --> 00:09:01,060
Notice that the curve
rises to infinity.
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00:09:01,060 --> 00:09:05,850
In other words, the maximum
value is undefined on this
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00:09:05,850 --> 00:09:07,090
particular closed interval.
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00:09:07,090 --> 00:09:09,470
You see there's a
jump over here.
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00:09:09,470 --> 00:09:12,290
Of course, the fact that this
is an infinite jump may make
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00:09:12,290 --> 00:09:14,330
you feel uneasy if
you don't feel
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00:09:14,330 --> 00:09:15,840
comfortable with infinity.
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00:09:15,840 --> 00:09:19,240
So let me paraphrase this just
a little bit in terms of a
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00:09:19,240 --> 00:09:20,370
finite jump.
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00:09:20,370 --> 00:09:22,790
Let me write down the
following function.
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00:09:22,790 --> 00:09:28,610
Let's think of the function 'f
of x' equals 'x' if 'x' is
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00:09:28,610 --> 00:09:29,840
less than 1.
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00:09:29,840 --> 00:09:32,610
Then as soon as 'x' is at
least as big as 1, the
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00:09:32,610 --> 00:09:34,620
function becomes 'minus x'.
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00:09:34,620 --> 00:09:37,380
In other words, graphically,
we have the line 'y' equals
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00:09:37,380 --> 00:09:41,670
'x' from 0 up to 1, but
not including 1.
186
00:09:41,670 --> 00:09:45,560
And then at 1, the function
jumps down to minus 1 and
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00:09:45,560 --> 00:09:48,310
becomes the line 'y'
equals 'minus x'.
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00:09:48,310 --> 00:09:50,800
Let's look, for example,
at the closed interval
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00:09:50,800 --> 00:09:53,370
again from 0 to 2.
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00:09:53,370 --> 00:09:56,390
What is the biggest value?
191
00:09:56,390 --> 00:10:00,590
What is the maximum that the
function can have if 'x' is
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00:10:00,590 --> 00:10:01,950
between 0 and 2?
193
00:10:01,950 --> 00:10:07,510
Well, notice that as 'x' gets
closer and closer to 1, 'f of
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00:10:07,510 --> 00:10:11,150
x' gets closer and closer to 1,
But 'f of x' never equals
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00:10:11,150 --> 00:10:14,880
exactly 1, because when
'x' is 1, the curve
196
00:10:14,880 --> 00:10:16,460
jumps down to here.
197
00:10:16,460 --> 00:10:20,050
In other words, notice that as
'x' gets arbitrarily close to
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00:10:20,050 --> 00:10:24,310
1, 'f of x' increases and gets
arbitrarily close to but
199
00:10:24,310 --> 00:10:26,150
never equals 1.
200
00:10:26,150 --> 00:10:29,580
In other words, notice that by
picking 'x' to be less than 1,
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00:10:29,580 --> 00:10:32,780
we can make 'f of x' as close
to 1 as we want, but we can
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00:10:32,780 --> 00:10:36,300
never make it exactly equal to
1 because of this jump which
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00:10:36,300 --> 00:10:37,980
takes place over here.
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00:10:37,980 --> 00:10:40,730
You see, this discontinuity
causes us a little bit of
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00:10:40,730 --> 00:10:42,270
difficulty.
206
00:10:42,270 --> 00:10:44,530
By the way, we can use
this example from
207
00:10:44,530 --> 00:10:46,140
another point of view.
208
00:10:46,140 --> 00:10:49,000
If we just look at the part
where it says 'f of x' is
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00:10:49,000 --> 00:10:52,040
equal to 'x' where 'x' is less
than 1, that means that we'll
210
00:10:52,040 --> 00:10:53,920
just look at this part
of the curve.
211
00:10:53,920 --> 00:10:57,410
Notice that this part of
the curve is unbroken.
212
00:10:57,410 --> 00:11:02,560
And yet as 'x' approaches 1, 'f
of x' approaches but never
213
00:11:02,560 --> 00:11:07,090
equals 1, which means that to
get the maximum value in here,
214
00:11:07,090 --> 00:11:10,170
I would have to include the
end point 'x' equals 1.
215
00:11:10,170 --> 00:11:12,150
In other words, notice that
even if the function is
216
00:11:12,150 --> 00:11:17,880
continuous, if the interval is
open, it may not assume the
217
00:11:17,880 --> 00:11:19,170
maximum value.
218
00:11:19,170 --> 00:11:21,950
In other words, you may not be
able to find the value of 'x'
219
00:11:21,950 --> 00:11:25,230
in that interval such as the
function will be maximum at
220
00:11:25,230 --> 00:11:26,650
that particular value.
221
00:11:26,650 --> 00:11:29,790
Now again, this is making a
mountain out of a mole hill in
222
00:11:29,790 --> 00:11:32,480
terms of your intuition because
these are result which
223
00:11:32,480 --> 00:11:34,945
I'm sure you believe
are true anyway.
224
00:11:34,945 --> 00:11:36,850
In other words, it
seems to be true.
225
00:11:36,850 --> 00:11:40,250
But all I hope is that these
little examples here show you
226
00:11:40,250 --> 00:11:43,470
what the importance of
continuity is, because as we
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00:11:43,470 --> 00:11:47,330
progress later in our work,
these so called fine points
228
00:11:47,330 --> 00:11:50,660
are going to become crucial
in many of our proofs.
229
00:11:50,660 --> 00:11:55,260
And by the way, I think it's a
truism to say that if you make
230
00:11:55,260 --> 00:11:58,970
fine points seem very, very
important before you get to
231
00:11:58,970 --> 00:12:02,190
use them and don't emphasize
them, people tend to think
232
00:12:02,190 --> 00:12:04,290
that you overestimated
your case.
233
00:12:04,290 --> 00:12:08,400
So I prefer not to beat this to
death and wait until such
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00:12:08,400 --> 00:12:11,340
times in our course that we need
these results before I
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00:12:11,340 --> 00:12:12,740
emphasize them more.
236
00:12:12,740 --> 00:12:15,190
So these are results which are
in the textbook, which I'll
237
00:12:15,190 --> 00:12:17,850
test you on in the exercises and
the like, but just want to
238
00:12:17,850 --> 00:12:21,580
go through with you now so that
you get an overall idea.
239
00:12:21,580 --> 00:12:24,360
Another result that's rather
clear geometrically is
240
00:12:24,360 --> 00:12:27,760
something called the
intermediate value theorem.
241
00:12:27,760 --> 00:12:30,730
Suppose that 'f' is continuous
on the closed interval from
242
00:12:30,730 --> 00:12:31,900
'a' to 'b'.
243
00:12:31,900 --> 00:12:34,740
And without loss of generality,
let's suppose 'f
244
00:12:34,740 --> 00:12:37,310
of a' is less than 'f of b'.
245
00:12:37,310 --> 00:12:39,920
The same kind of an argument
would hold if 'f of b' were
246
00:12:39,920 --> 00:12:42,400
less than 'f of a', but I just
need some sort of an
247
00:12:42,400 --> 00:12:43,600
orientation.
248
00:12:43,600 --> 00:12:47,830
Now I say let 'm' be any number
such that m is what?
249
00:12:47,830 --> 00:12:49,440
Between 'f of a' and 'f of b'.
250
00:12:49,440 --> 00:12:51,900
In other words, 'm' is greater
than 'f of a' but
251
00:12:51,900 --> 00:12:53,540
less than 'f of b'.
252
00:12:53,540 --> 00:12:56,210
You see, if 'f of b' were less
than 'f of a', I would just
253
00:12:56,210 --> 00:12:58,470
reverse the inequality
signs here.
254
00:12:58,470 --> 00:13:01,600
At any rate, what my claim is
that we can find the number
255
00:13:01,600 --> 00:13:05,770
'c' in the open interval from
'a' to 'b' such that 'f of c'
256
00:13:05,770 --> 00:13:07,610
equals 'm'.
257
00:13:07,610 --> 00:13:10,110
Again, this may sound
kind of complicated.
258
00:13:10,110 --> 00:13:13,430
All it says in terms of
a picture is this.
259
00:13:13,430 --> 00:13:20,780
If our curve goes from point 1
to point 2 where the height of
260
00:13:20,780 --> 00:13:25,170
'p1' is 'f of a' and the height
of 'p2' is 'f of b',
261
00:13:25,170 --> 00:13:29,680
then it must assume every
intermediary height between 'f
262
00:13:29,680 --> 00:13:33,890
of a' and 'f of b' on that
particular interval.
263
00:13:33,890 --> 00:13:36,810
Again, you can think of
that in terms of an
264
00:13:36,810 --> 00:13:38,250
automobile, if you want.
265
00:13:38,250 --> 00:13:42,160
If an automobile goes from a
speed of 20 miles an hour to a
266
00:13:42,160 --> 00:13:46,000
speed of 30 miles per hour in
what we think of as being
267
00:13:46,000 --> 00:13:50,550
continuous motion, then at some
time in that interval, it
268
00:13:50,550 --> 00:13:54,640
must've had any particular
speed you want to mention
269
00:13:54,640 --> 00:13:56,080
between 20 and 30 miles.
270
00:13:56,080 --> 00:13:58,620
In other words, we don't
visualize a car going from 20
271
00:13:58,620 --> 00:14:02,430
miles an hour to 30 miles an
hour, say, without at one time
272
00:14:02,430 --> 00:14:05,430
having passed the speed
of 27 miles per hour
273
00:14:05,430 --> 00:14:06,850
or some such thing.
274
00:14:06,850 --> 00:14:10,110
And geometrically, you see,
this proof is very simple,
275
00:14:10,110 --> 00:14:12,840
namely, draw the line
'y' equals 'm'.
276
00:14:12,840 --> 00:14:16,460
277
00:14:16,460 --> 00:14:19,410
And now what you're saying is
you want to get from this
278
00:14:19,410 --> 00:14:23,680
point to this point.
279
00:14:23,680 --> 00:14:26,320
And the idea is that the only
way you can get from this
280
00:14:26,320 --> 00:14:30,120
point to this point without
crossing this line is if
281
00:14:30,120 --> 00:14:33,430
someplace along the way, you
jump over this line.
282
00:14:33,430 --> 00:14:35,750
In other words, geometrically,
all you're saying is lookit,
283
00:14:35,750 --> 00:14:42,520
if I have to join 'p1' and 'p2'
with an unbroken line,
284
00:14:42,520 --> 00:14:46,010
then I must cross the line
'y' equals 'm' someplace.
285
00:14:46,010 --> 00:14:49,820
By the way, notice I say that at
least one place, the curve
286
00:14:49,820 --> 00:14:53,040
could have done something
like this, for example.
287
00:14:53,040 --> 00:14:54,200
But the important
point is what?
288
00:14:54,200 --> 00:14:56,870
There's at least one number
'c' such that 'f
289
00:14:56,870 --> 00:14:59,230
of c' equals 'm'.
290
00:14:59,230 --> 00:15:02,830
Again, somebody might think in
terms of an end run and say
291
00:15:02,830 --> 00:15:05,830
couldn't we have done something
like this and not
292
00:15:05,830 --> 00:15:07,070
cross this line?
293
00:15:07,070 --> 00:15:10,570
That brings us back again
to our concept of
294
00:15:10,570 --> 00:15:12,060
single-valuedness.
295
00:15:12,060 --> 00:15:15,500
In other words, notice, if we
remove the restriction that 'f
296
00:15:15,500 --> 00:15:18,560
be a' single-valued function,
in other words, if 'f' can
297
00:15:18,560 --> 00:15:22,590
double back, notice that if we
think of the line 'y' equals
298
00:15:22,590 --> 00:15:27,060
'm' as being endless, the
point remains what?
299
00:15:27,060 --> 00:15:30,420
That you must someplace cross
this line in going
300
00:15:30,420 --> 00:15:33,280
from 'p1' to 'p2'.
301
00:15:33,280 --> 00:15:36,760
But if we remove the
single-valued restriction,
302
00:15:36,760 --> 00:15:39,230
then the point at which
you cross this line
303
00:15:39,230 --> 00:15:41,690
would not have to be--
304
00:15:41,690 --> 00:15:46,020
see, in this case, 'c' would
not be in the open interval
305
00:15:46,020 --> 00:15:54,610
from 'a' to 'b' since 'f'
is not single- valued.
306
00:15:54,610 --> 00:15:57,350
And what I'm hoping this
discussion does for us is
307
00:15:57,350 --> 00:16:00,460
gives us a good geometric
feeling as to what is
308
00:16:00,460 --> 00:16:04,360
happening in terms of a
continuous function.
309
00:16:04,360 --> 00:16:07,410
By the same token, remember
our basic definition of
310
00:16:07,410 --> 00:16:11,180
continuity goes back to
our concept of limit.
311
00:16:11,180 --> 00:16:14,300
And our concept of limit has
been cemented down fairly
312
00:16:14,300 --> 00:16:17,430
firmly from an analytical point
of view, and that means
313
00:16:17,430 --> 00:16:20,410
that we can also see these
things theoretically as well
314
00:16:20,410 --> 00:16:21,590
as pictorially.
315
00:16:21,590 --> 00:16:25,570
And hopefully what we will do
is combine pictorial and
316
00:16:25,570 --> 00:16:27,930
analytic aspects to
the best possible
317
00:16:27,930 --> 00:16:30,000
advantage for solving problems.
318
00:16:30,000 --> 00:16:32,780
But to show you what I mean by
this, let's look at a few
319
00:16:32,780 --> 00:16:36,310
analytical properties of
continuous functions.
320
00:16:36,310 --> 00:16:41,560
For example, suppose 'f' and 'g'
are both continuous at 'x'
321
00:16:41,560 --> 00:16:42,710
equals 'a'.
322
00:16:42,710 --> 00:16:46,290
And suppose we define 'h' to
be the sum of 'f' and 'g'.
323
00:16:46,290 --> 00:16:50,730
In other words, 'h of x' is 'f
of x' plus 'g of x', OK?
324
00:16:50,730 --> 00:16:54,010
Let's compute the limit of 'h
of x' as 'x' approaches 'a'.
325
00:16:54,010 --> 00:16:57,410
By definition, the limit of 'h
of x' as 'x' approaches 'a' is
326
00:16:57,410 --> 00:17:00,630
the limit of the quantity 'f
of x' plus 'g of x' as 'x'
327
00:17:00,630 --> 00:17:01,870
approaches 'a'.
328
00:17:01,870 --> 00:17:07,230
Now, if we look at this
particular expression, notice
329
00:17:07,230 --> 00:17:10,579
that because of our theorems on
limits, we can do an awful
330
00:17:10,579 --> 00:17:13,819
lot with this without having
to make any recourse to our
331
00:17:13,819 --> 00:17:14,849
picture at all.
332
00:17:14,849 --> 00:17:18,099
For example, notice that we can
say right away that since
333
00:17:18,099 --> 00:17:20,220
the limit of a sum is the sum
of the limits, this is the
334
00:17:20,220 --> 00:17:22,910
limit of 'f of x' as 'x'
approaches 'a' plus the limit
335
00:17:22,910 --> 00:17:25,290
of 'g of x' as 'x'
approaches 'a'.
336
00:17:25,290 --> 00:17:29,170
Secondly, since 'f' and 'g' are
continuous at 'x' equals
337
00:17:29,170 --> 00:17:31,930
'a', by definition of continuous
that 'x' equals
338
00:17:31,930 --> 00:17:34,700
'a', this says that the limit of
'f of x' as 'x' approaches
339
00:17:34,700 --> 00:17:35,990
'a' is 'f of a'.
340
00:17:35,990 --> 00:17:39,410
The limit of 'g of x' as 'x'
approaches 'a' is 'g of a'.
341
00:17:39,410 --> 00:17:42,690
Therefore, this expression is
'f of a' plus 'g of a'.
342
00:17:42,690 --> 00:17:46,960
But by the definition of 'h',
this is just 'h of a'.
343
00:17:46,960 --> 00:17:50,340
And if we now look at
this, we see what?
344
00:17:50,340 --> 00:17:53,730
That the limit of 'h of x'
as 'x' approaches 'a'
345
00:17:53,730 --> 00:17:55,250
equals 'h of a'.
346
00:17:55,250 --> 00:17:59,090
Notice that this result came
about under the assumption
347
00:17:59,090 --> 00:18:01,760
that 'f' and 'g' were both
continuous that 'x' equals
348
00:18:01,760 --> 00:18:04,060
'a', and what have we
proven over here?
349
00:18:04,060 --> 00:18:06,470
This is precisely the statement
that 'h' is
350
00:18:06,470 --> 00:18:08,680
continuous at 'x' equals 'a'.
351
00:18:08,680 --> 00:18:11,500
In other words, what we've now
proven analytically is that a
352
00:18:11,500 --> 00:18:13,640
sum of two continuous
functions is
353
00:18:13,640 --> 00:18:15,560
a continuous function.
354
00:18:15,560 --> 00:18:18,470
By induction, we could prove
this for the sum
355
00:18:18,470 --> 00:18:19,880
of more than two.
356
00:18:19,880 --> 00:18:23,160
We can in a similar way by using
limit theorems prove
357
00:18:23,160 --> 00:18:25,898
things like the product of two
continuous functions is
358
00:18:25,898 --> 00:18:28,300
continuous, et cetera.
359
00:18:28,300 --> 00:18:31,690
We also were on the verge of a
topic like this when we talked
360
00:18:31,690 --> 00:18:34,520
about differentiability
sometime back.
361
00:18:34,520 --> 00:18:36,990
Namely, I claim that there is
an interesting connection
362
00:18:36,990 --> 00:18:39,690
between differentiable and
continuous, and that the
363
00:18:39,690 --> 00:18:43,070
connection is that any
differentiable function is
364
00:18:43,070 --> 00:18:44,220
continuous.
365
00:18:44,220 --> 00:18:48,020
And by the way, the proofs
utilizes that which we used in
366
00:18:48,020 --> 00:18:51,040
our first lecture on derivatives
in this block.
367
00:18:51,040 --> 00:18:54,200
Namely, we take the expression
'f' of--
368
00:18:54,200 --> 00:18:58,370
we want to show that if 'f prime
of a' exists, that 'f'
369
00:18:58,370 --> 00:19:00,690
must be continuous at
'x' equals 'a'.
370
00:19:00,690 --> 00:19:03,550
That means we want to show that
the limit of 'f of x' as
371
00:19:03,550 --> 00:19:06,240
'x' approaches 'a' is 'f of
a', and that's the same as
372
00:19:06,240 --> 00:19:09,550
saying we want to show that this
difference approaches 0
373
00:19:09,550 --> 00:19:10,360
and the limit.
374
00:19:10,360 --> 00:19:14,320
And the trick is we take 'f of
x' minus 'f of a' and write it
375
00:19:14,320 --> 00:19:20,230
as ''f of x' minus 'f of a'
divided by 'x - a'' times 'x -
376
00:19:20,230 --> 00:19:23,270
a', the idea being that we'll
now take the limit as 'x'
377
00:19:23,270 --> 00:19:24,130
approaches 'a'.
378
00:19:24,130 --> 00:19:27,760
The limit of a product is the
product of the limits.
379
00:19:27,760 --> 00:19:30,360
The limit of this as 'x'
approaches 'a' is just going
380
00:19:30,360 --> 00:19:31,600
to be 'f prime of a'.
381
00:19:31,600 --> 00:19:33,390
That's our definition
of derivative.
382
00:19:33,390 --> 00:19:37,090
And the limit of 'x - a' as 'x'
approaches 'a' is just 0.
383
00:19:37,090 --> 00:19:38,920
In other words, putting
these steps together,
384
00:19:38,920 --> 00:19:39,870
look what we get.
385
00:19:39,870 --> 00:19:43,820
The limit as 'x' approaches 'a',
'f of x' minus 'f of a',
386
00:19:43,820 --> 00:19:47,010
is the limit as 'x' approaches
'a' of this bracketed
387
00:19:47,010 --> 00:19:51,400
expression ''f of x' minus 'f of
a' over 'x - a'' times the
388
00:19:51,400 --> 00:19:54,340
limit of 'x - a' as 'x'
approaches 'a'.
389
00:19:54,340 --> 00:19:58,420
By definition, 'f' being
differentiable at 'a' means
390
00:19:58,420 --> 00:20:02,430
that this limit exists and is,
in fact, 'f prime of a'.
391
00:20:02,430 --> 00:20:06,530
But any finite number
times 0 is still 0.
392
00:20:06,530 --> 00:20:09,350
And therefore, it follows that
the limit of 'f of x' as 'x'
393
00:20:09,350 --> 00:20:11,150
approaches 'a' is 'f of a'.
394
00:20:11,150 --> 00:20:16,780
There's a legitimate analytic
proof that differentiability
395
00:20:16,780 --> 00:20:18,650
implies continuity.
396
00:20:18,650 --> 00:20:21,760
Now, it's not necessarily true
that continuity implies
397
00:20:21,760 --> 00:20:23,190
differentiability.
398
00:20:23,190 --> 00:20:25,720
And you see even though you
can prove these things
399
00:20:25,720 --> 00:20:28,370
analytically, if you don't
form some picture in your
400
00:20:28,370 --> 00:20:32,720
mind, you frequently will my
wind up memorizing results
401
00:20:32,720 --> 00:20:34,890
rather than having a
feeling for them.
402
00:20:34,890 --> 00:20:39,870
What I mean by having pictures
and proofs coexist is this.
403
00:20:39,870 --> 00:20:42,570
Instead of saying to yourself,
let's see, continuity is a
404
00:20:42,570 --> 00:20:44,550
weaker condition than
differentiability, it's a
405
00:20:44,550 --> 00:20:45,890
strong-- which one is it?
406
00:20:45,890 --> 00:20:47,520
You say look-it.
407
00:20:47,520 --> 00:20:50,880
We identified continuity
with unbroken.
408
00:20:50,880 --> 00:20:53,230
Remember, continuous
means unbroken.
409
00:20:53,230 --> 00:20:55,380
What did differentiable mean?
410
00:20:55,380 --> 00:20:58,480
Differentiable meant that the
curve had a tangent line, and
411
00:20:58,480 --> 00:21:01,260
that, in turn, meant that
the curve was smooth.
412
00:21:01,260 --> 00:21:04,290
In other words, we may think
of differentiable as the
413
00:21:04,290 --> 00:21:07,960
geometric analog of smoothness,
continuity as the
414
00:21:07,960 --> 00:21:10,740
geometric analog of
unbrokenness.
415
00:21:10,740 --> 00:21:13,990
And now I think it's very easy
to see pictorially that a
416
00:21:13,990 --> 00:21:18,430
smooth curve must be unbroken,
but an unbroken curve doesn't
417
00:21:18,430 --> 00:21:19,630
have to be smooth.
418
00:21:19,630 --> 00:21:22,540
And in fact, a very trivial
illustration of this is the
419
00:21:22,540 --> 00:21:25,900
graph 'y' equals the absolute
value of 'x'.
420
00:21:25,900 --> 00:21:27,740
You see, well, what
happens here?
421
00:21:27,740 --> 00:21:32,370
At the origin, notice that we
have a sharp corner, but that
422
00:21:32,370 --> 00:21:34,030
the curve itself is
unbroken there.
423
00:21:34,030 --> 00:21:37,080
You see, I can draw the curve
without taking the pencil off
424
00:21:37,080 --> 00:21:38,550
the paper, the chalk
off the board.
425
00:21:38,550 --> 00:21:40,340
That's what continuity
means from an
426
00:21:40,340 --> 00:21:41,910
intuitive point of view.
427
00:21:41,910 --> 00:21:43,900
Yet I have a sharp corner.
428
00:21:43,900 --> 00:21:45,680
It's not smooth here.
429
00:21:45,680 --> 00:21:48,030
That's why I write smooth
in quotation marks.
430
00:21:48,030 --> 00:21:52,490
Hopefully, the textbook,
together with our exercises,
431
00:21:52,490 --> 00:21:54,050
will make this much
clear to you.
432
00:21:54,050 --> 00:21:57,350
But notice again in terms of
the picture how an unbroken
433
00:21:57,350 --> 00:22:00,450
curve doesn't have to be smooth,
but a smooth curve has
434
00:22:00,450 --> 00:22:01,410
to be unbroken.
435
00:22:01,410 --> 00:22:04,180
Now, the point is we
could say much more
436
00:22:04,180 --> 00:22:05,830
about continuous functions.
437
00:22:05,830 --> 00:22:08,580
I think that all we have
to say, though, has
438
00:22:08,580 --> 00:22:09,770
already been said.
439
00:22:09,770 --> 00:22:12,180
It'll be reinforced
in the text.
440
00:22:12,180 --> 00:22:13,870
We'll have exercises on this.
441
00:22:13,870 --> 00:22:16,810
But the important thing for now
is to understand what we
442
00:22:16,810 --> 00:22:20,605
mean by continuous, how we will
use it in the future, and
443
00:22:20,605 --> 00:22:23,860
at any rate, we'll be talking
about that more later.
444
00:22:23,860 --> 00:22:25,640
And so until next
time, goodbye.
445
00:22:25,640 --> 00:22:28,450
446
00:22:28,450 --> 00:22:30,980
NARRATOR: Funding for the
publication of this video was
447
00:22:30,980 --> 00:22:35,700
provided by the Gabriella and
Paul Rosenbaum Foundation.
448
00:22:35,700 --> 00:22:39,870
Help OCW continue to provide
free and open access to MIT
449
00:22:39,870 --> 00:22:44,070
courses by making a donation
at ocw.mit.edu/donate.
450
00:22:44,070 --> 00:22:48,810