Take the 2-minute tour ×  While studying complex variables, I
could learn that $f(z)=|z|^{2}$ has only one point which is
$z=0$ that $f$ being differentiable and $f$ being not
differentiable at any other points.  Then, I was wondering if
there is a function $f: \mathbb R \to \mathbb R$ that has only
one point differentiable and not on any other points.  In
intuition, it seems there are no such point! However, I have no
idea how I can prove this...  Additional question is that would
there be any function $f: \mathbb R \to \mathbb R$ that has only
one point continuous and not on any other points.  I think this
is pretty interesting things to think about! :-)  share|improve
this question Here's a thread on continuous functions
differentiable only at one point. –  t.b. Sep 11 '12 at 15:36
Note that André's example below is a function that is only
continuous at one point and is only differentiable at one point.
–  copper.hat Sep 11 '12 at 15:41 If you like such questions,
you might want to check out the books by B