Question

Let $f : R \rightarrow R$ be defined as $f(x) = \begin{cases} 0, & \text{ x is irrational} \\[2ex] sin | x |, & \text{x is rational } \end{cases}$
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, $f(x) = \begin{cases} 0, & \text{ x is irrational} \\[2ex] \sin | x |, & \text{x is rational } \end{cases}$
If $f(x)$ is continuous, then $\sin |x|=0$
$\Rightarrow x=k \pi$, where $k$ is an integer.