1. Tardigrade
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  3. Mathematics
  4. Let f(x) = begincases x, textif x text is irrational 0 textif x text is rational endcases then f is

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Q. Let $ f(x) = \begin{cases} x, & \quad \text{if } x \text{ is irrational}\\ 0 & \quad \text{if } x \text{ is rational}\\ \end{cases} $
then $f$ is

KCETKCET 2013Continuity and Differentiability

Solution:

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