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(D) The function is given by $f(x) = [a + 13 \sin x]$ for $x \in (0, \pi)$.Since $a$ is an integer,we can write $f(x) = a + [13 \sin x]$.The function

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$25$

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(D) The function is given by $f(x) = [a + 13 \sin x]$ for $x \in (0, \pi)$.Since $a$ is an integer,we can write $f(x) = a + [13 \sin x]$.The function $f(x)$ is not differentiable at points where the argument of the greatest integer function,$13 \sin x$,is an integer.For $x \in (0, \pi)$,the range of $13 \sin x$ is $(0, 13]$.The values of $13 \sin x$ are integers at $13 \sin x = 1, 2, 3, \dots, 13$.For each integer $k \in \{1, 2, \dots, 12\}$,there are $2$ values of $x$ in $(0, \pi)$ such that $13 \sin x = k$.For $k = 13$,there is only $1$ value of $x$ in $(0, \pi)$,which is $x = \frac{\pi}{2}$.Thus,the total number of points where the function is not differentiable is $2 	imes 12 + 1 = 25$.

Let $a \in Z$ and $[t]$ be the greatest integer $\leq t$. Then the number of points,where the function $f(x) = [a + 13 \sin x], x \in (0, \pi)$ is not differentiable,is $........$.

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