Question

Let $f(x)=\{\begin{array}{ll}x,& \text{if }x\text{ is irrational}\\ 0& \text{if }x\text{ is rational}\end{array}$
then $f$ is

• A. continuous everywhere
• B. discontinuous everywhere
• C. continuous only at $x=0$
• D. continuous at all rational numbers

Answer 1

Answer: (C)

Given, $f(x)=\{\begin{array}{ll}x,& \text{ if }x\text{ is irrational }\\ 0,& \text{ if }x\text{ is rational }\end{array}$
$LHL=\underset{x\to 0^{-}}{\lim }f(x)=\underset{x\to 0^{-}}{\lim }x=0$
$RHL=\underset{x\to 0^{+}}{\lim }f(x)=\underset{x\to 0^{+}}{\lim }x=0$
and $f(0)=0$
Hence, $f(x)$ is continuous at $x=0$.