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 $........$.

If $f(x) = \begin{cases} x^2 & \text{if } x \leqslant x_0 \\ ax + b & \text{if } x > x_0 \end{cases}$ is derivable for all $x \in \mathbb{R}$,then the values of $a$ and $b$ are respectively:

The number of points at which the function,$f(x) = |x - 0.5| + |x - 1| + \tan x$ does not have a derivative in the interval $(0, 2)$ is :

Prove that the function $f$ given by $f(x) = |x - 1|, x \in R$ is not differentiable at $x = 1$.

Consider the following statements.
$(a)$ If a function is differentiable at a point $p$ then it is not continuous at $p$.
$(b)$ If a function is not continuous at $x = a$, then it is not differentiable at $x = a$.
$(c)$ If $f(x) = |x|$ then $f(x)$ is not differentiable but continuous on $R$.
$(d)$ If $f(x) = x - [x]$, then $f'(1) = 1$.
Which of the above statements are (is) correct?

If $f(x) = |x - 3|,$ then $f$ is