Differentiability only in isolated point

Question

Do functions exist, which are differentiable in a point, but not in a neighborhood of this point?

Is $e^{\frac{1}{W(x)-2}}$, where W is the Weierstrass function, maybe an example of a such function?

I think $x^2$ times the Dirichlet function D(x) has this property at x=0 — Maxim Umansky, Jul 16 '13 at 04:32
I finally remembered this old math joke your name alludes to: what's yellow, normed, and complete? — Julien, Jul 16 '13 at 07:22

score 8 · Answer 1 · edited Jul 16 '13 at 07:15

8

The function $$f(x)=\chi_{\Bbb Q}(x)\cdot x^2$$ is differentiable (and continuous) only at $x=0$.

edited Jul 16 '13 at 07:15

Did

answered Jul 16 '13 at 04:21

Pedro

Simplified your example. Please revert if you disagree. – Did Jul 16 '13 at 07:15
@Did I don't. In fact, what I wrote before was unnecessary. – Pedro Jul 16 '13 at 07:18
@Did Could you shed some light - if possible - on robjohn's counterexample in the comments? – Pedro Jul 16 '13 at 07:20
The solution in the book jumps from [$(v_n)$ converges to $v$] to [$f(v)=v$]. This step only holds if $f$ is continuous (at $v$). Hence, either the assumption that $f$ is continuous is in the book, or it was forgotten. – Did Jul 16 '13 at 08:35
Oh, and +1 for the solution here. – Did Jul 16 '13 at 08:36
@Did Ah, thank you on both things, but I'm capped! – Pedro Jul 16 '13 at 08:36

score 6 · Accepted Answer · edited Jul 16 '13 at 04:26

6

If we multiply the Weierstrass function by $x^2$, we get a function which is continuous everywhere, and differentiable at $0$ but nowhere else.

edited Jul 16 '13 at 04:26

Pedro

answered Jul 16 '13 at 04:20

André Nicolas

Thanks for all responses. I set this answer as my accepted answer because I was looking for a continuous function. Sorry for not being clear. – Bananach Jul 16 '13 at 13:43

2 Answers2