]> 6.3 Examples of non Differentiable Behavior

## 6.3 Examples of non Differentiable Behavior

A function which jumps is not differentiable at the jump nor is one which has a cusp, like $| x |$ has at $x = 0$ .

Generally the most common forms of non-differentiable behavior involve a function going to infinity at $x$ , or having a jump or cusp at $x$ .

There are however stranger things. The function $sin ⁡ ( 1 x )$ , for example is singular at $x = 0$ even though it always lies between -1 and 1. Its hard to say what it does right near 0 but it sure doesn't look like a straight line.

If the function $f$ has the form $f = g h , f$ will usually be singular at argument $x$ if $h$ vanishes there, $h ( x ) = 0$ . However if $g$ vanishes at $x$ as well, then $f$ will usually be well behaved near $x$ , though strictly speaking it is undefined there.

We usually define $f$ at $x$ under such circumstances to be the ratio of the linear approximation at $x$ to g to that to $h$ very near $x$ , which means we define $f ( x )$ to be $f = g ' ( x ) h ' ( x )$ , when, of course the denominator here does not vanish. (If the denominator does vanish and the numerator vanishes as well, you can try to define $f ( x )$ similarly as the ratio of the derivatives of these derivatives, etc.)

This kind of thing, an isolated point at which a function is not defined, is called a "removable singularity" and the procedure for removing it just discussed is called "l' Hospital's rule" .

An example is $sin ⁡ x x$ at $x = 0$ .

Continuous but non differentiable functions