A function which jumps is not differentiable at the jump nor is one which has a cusp, like
$\left|x\right|$
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
$\mathrm{sin}\left(\frac{1}{x}\right)$
, 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
$f=\frac{g}{h},f$
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
$f=\frac{g\text{'}(x)}{h\text{'}(x)}$
, 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
$\frac{\mathrm{sin}x}{x}$
at
$x=0$
.