Question

Let $f:R\to R$ be defined as
$f\left(x\right)=\{\begin{array}{cc}0,& \text{x is irrational}\\ \sin |x|,& \text{x is rational}\end{array}$

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

The correct option is D $f$ is continuous at $x=k\pi$, where $k$ is an integer.

For every rational number $x$, we can find an irrational number very close to the left or right of $x$.
So, $f$ is continuous at $x$ if and only if
$\sin |x|=0$
$\Rightarrow x=k\pi$