Question

Let $f:R\to R$ be defined as $f(x)=\{\begin{array}{ll}0,& \text{ x is irrational}\\ sin|x|,& \text{x is rational }\end{array}$
Then which of the following is true ?

• A. $f$ is discontinuous for all $x$
• B. $f$ is continuous for all $x$
• C. $f$ is discontinuous at $x=k\pi$, where $k$ is an integer
• D. $f$ is continuous at $x=k\pi$, where $k$ is an integer

Answer 1

Answer: (D)

we have, $f(x)=\{\begin{array}{ll}0,& \text{ x is irrational}\\ \sin |x|,& \text{x is rational }\end{array}$
If $f(x)$ is continuous, then $\sin |x|=0$
$\Rightarrow x=k\pi$, where $k$ is an integer.