TITLE: Differentiability only in isolated point
QUESTION [6 upvotes]: 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?

REPLY [8 votes]: The function $$f(x)=\chi_{\Bbb Q}(x)\cdot x^2$$ is differentiable (and continuous) only at $x=0$.

REPLY [6 votes]: If we multiply the Weierstrass function by $x^2$, we get a function which is continuous everywhere, and differentiable at $0$ but nowhere else.