Basic Analysis I — Real Analysis · Problem 3.2.3

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · continuous-functions

Problem

Define

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

f(x) := \begin{cases} x & \text{if } x \text{ is rational,} \\ x^2 & \text{if } x \text{ is irrational.} \end{cases}

Using the definition of continuity directly prove that

f

is continuous at

1

and discontinuous at

2