Diffentiable function only at zero

Question

Is there a case where $f$ is differentiable at $0$ but not any other point?

We might want to put stronger conditions on the function, like continuity. Then take any of the standard continuous nowhere differentiable functions $g(x)$, and let $f(x)=x^2g(x)$. — André Nicolas, Nov 15 '15 at 22:43

score 2 · Answer 1 · answered Nov 15 '15 at 22:36

2

Yes this is possible. Consider the function $f= x^2$ when $x \in \mathbb Q$ and $-x^2$ otherwise.

This map is continuous only at 0, and in fact can be shown using a simple limit argument to also be differentiable there!

answered Nov 15 '15 at 22:36

A. Thomas Yerger

score 2 · Answer 2 · answered Nov 15 '15 at 22:36

2

Try $$f(x)=\begin{cases}x^2&\text{if }x\in\Bbb Q\\0&\text{otherwise}\end{cases}$$

answered Nov 15 '15 at 22:36

Hagen von Eitzen

2 Answers2