If the function $f(x) = \begin{cases} \frac{(e^{kx}-1) \sin kx}{4 \tan x}, & x \neq 0 \\ P, & x=0 \end{cases}$ is differentiable at $x=0$, then

If $f(x) = \begin{cases} 2a - x & \text{when } -a < x < a \\ 3x - 2a & \text{when } a \leq x \end{cases}$,then which of the following is true?

The function given by $y = ||x| - 1|$ is differentiable for all real numbers except the points

If $f(x) = [x]$,where $[x]$ is the greatest integer not greater than $x$,then $f^{\prime}(1^{+}) = \dots$.

If $f(x) = \begin{cases} ax^2 + b; & x \le 0 \\ x^2; & x > 0 \end{cases}$ possesses a derivative at $x = 0$,then:

Let the functions $f, g$ and $h$ be defined as follows:
$f(x) = \begin{cases} x \sin \left( \frac{1}{x} \right) & \text{for } -1 \le x \le 1, x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$
$g(x) = \begin{cases} x^2 \sin \left( \frac{1}{x} \right) & \text{for } -1 \le x \le 1, x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$
$h(x) = |x|^3$ for $-1 \le x \le 1$.
Which of these functions are differentiable at $x = 0$?