TITLE: Is the domain of a function necessarily the same as that of its derivative?
QUESTION [9 upvotes]: As in the title, I was wondering if it is necessarily true that the domain of a function is shared by its derivative.

REPLY [4 votes]: A more extreme example: the Weierstrass function
$$f(x) = \sum_{n=1}^{\infty}a^n\cos(b^n\pi x)$$
($0<a<1$, $b$ positive odd integer, $ab > 1 + 3\pi/2$)
has
$$\text{dom}(f) = \Bbb R,\qquad\text{dom}(f') = \emptyset.$$