Given function

$f(x) = [a + 13\sin x], \qquad x \in (0,\pi)$

where $[\cdot]$ denotes the greatest integer function.

When is $f(x)$ not differentiable?

The greatest integer function $[y]$ is not differentiable exactly when its argument $y$ is an integer.

So $f(x)$ is not differentiable at those $x \in (0,\pi)$ for which

$a + 13\sin x \in \mathbb{Z}.$

Use the fact that $a \in \mathbb{Z}$

Since $a$ is an integer, the condition

$a + 13\sin x \in \mathbb{Z}$

is equivalent to

$13\sin x \in \mathbb{Z}.$

Let

$13\sin x = n, \quad n \in \mathbb{Z}.$

Since $x \in (0,\pi)$, we have

$0 < \sin x \le 1.$

Hence

$0 < \frac{n}{13} \le 1 \implies n = 1,2,3,\dots,13.$

Count solutions for each integer $n$

We need the number of solutions of

$\sin x = \frac{n}{13}, \qquad x \in (0,\pi).$

• If $0 < \frac{n}{13} < 1$, i.e. $n=1,2,\dots,12$, then in $(0,\pi)$ the equation has two solutions: $x = \sin^{-1}\left(\frac{n}{13}\right), \qquad x = \pi - \sin^{-1}\left(\frac{n}{13}\right).$

• If $\frac{n}{13} = 1$, i.e. $n=13$, then $\sin x = 1$ has exactly one solution in $(0,\pi)$, namely $x = \frac{\pi}{2}.$

Total number of non-differentiable points

Therefore total number of points is

$12 \times 2 + 1 = 24 + 1 = 25.$

Final answer

$\boxed{25}$

Comparison with stored answer

Stored correct answer = $25$.

This matches the derived answer.