Let a ∈ ℤ and [t] be the greatest integer ≤ t , then the number of...

Given,

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

$⇒ f ( x ) = [ 13\sin x ] + a$, as $a \in \text{Integer}$

Now let $g ( x ) = 13\sin x$, now plotting the diagram of $y = 13\sin x$ we get,

Now from above diagram we can see that,

$f ( x ) =$\left[13\sin x\right]$is not differentiable at intersection points of$y = k, k \in [ 1, 13 ] \text{ and } k \in Z$</p><p>$⇒$Points of non-differentiability$= 12 \times 2 + 1 = 25