Q85 / 180 · KCET 2013 · Mathematics

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

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

Solution

Given, $f(x)= \begin{cases}x, & \text { if } x \text { is irrational } \\ 0, & \text { if } x \text { is rational }\end{cases}$ $\begin{aligned} &\mathrm{LHL}=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}} x=0 \\ &\mathrm{RHL}=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} x=0 \end{aligned}$ and $f(0)=0$ Hence, $f(x)$ is continuous at $x=0$.

Hence, the correct answer is option: 'C'.

Relevant Questions