Basic Analysis I — Real Analysis · Problem 4.1.5

The Derivative — Jiri Lebl

Mathematics — Real Analysis · differentiability-pathological-function

Problem

Define

f : \mathbb{R} \to \mathbb{R}

f(x) \mathrel{\mathop:}= \begin{cases} x^2 & \text{if } x \in \mathbb{Q}, \\ 0 & \text{otherwise.} \end{cases}

Prove that

f

is differentiable at

0

, but discontinuous at all points except

0