Question

Let $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{cc}0,& xis\text{Irrational}\\ \sin |x|,& xis\text{Rational}\end{array}$. 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 interger
• D. $f$ is continuous at $x=k\pi$, where $k$ is an integer

Answer 1

Answer: (D)

$f$ is continuous at $x=k\pi$, where $k$ is an integer

We have,

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