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(D) function $f(x)$ defined as $f(x) = \begin{cases} g(x), & x \in \mathbb{Q} \ h(x), & x 
otin \mathbb{Q} \end{cases}$ is continuous at $x = a$ if

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

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(D) function $f(x)$ defined as $f(x) = \begin{cases} g(x), & x \in \mathbb{Q} \ h(x), & x 
otin \mathbb{Q} \end{cases}$ is continuous at $x = a$ if and only if $g(a) = h(a)$. Here, $g(x) = \sin |x|$ and $h(x) = 0$. For continuity, we require $\sin |x| = 0$. This occurs when $|x| = n\pi$ for some integer $n$, which implies $x = n\pi$ for $n \in \mathbb{Z}$. At any point $x = k\pi$ (where $k \in \mathbb{Z}$), $f(x) = \sin |k\pi| = 0$. For any other point $x 
eq k\pi$, $\sin |x| 
eq 0$, so the function is discontinuous because the values for rational and irrational numbers do not coincide. Thus, $f$ is continuous only at $x = k\pi$ and discontinuous everywhere else. Therefore, option $D$ is correct.

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

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