# Differentiability only in isolated point

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 4:32
• I finally remembered this old math joke your name alludes to: what's yellow, normed, and complete? – Julien Jul 16 '13 at 7:22

If we multiply the Weierstrass function by $x^2$, we get a function which is continuous everywhere, and differentiable at $0$ but nowhere else.
The function $$f(x)=\chi_{\Bbb Q}(x)\cdot x^2$$ is differentiable (and continuous) only at $x=0$.
• 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 8:35