KCET · Maths · Continuity and Differentiability
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
- C continuous only at \(x=0\)
- D continuous at all rational numbers
Answer & Solution
Correct Answer
(C) continuous only at \(x=0\)
Step-by-step Solution
Detailed explanation
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 \(x=0\).
\(\begin{aligned}
&\mathrm{LHL}=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}} x=0 \\
&\mathrm{RHL}=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} x=0
\end{aligned}\)
and \(f(0)=0\) Hence, \(f(x)\) is continuous at \(x=0\).
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