The function which is continuous for all real values of $x$ and differentiable at $x = 0$ is

The function $y = \sin^{-1}(\cos x)$ is not differentiable at . . . . . .

For the function $f(x) = e^{\sin |x|} - |x|$, $x \in R$, consider the following statements:
Statement $I$: $f$ is differentiable for all $x \in R$.
Statement $II$: $f$ is increasing in $(-\pi, -\frac{\pi}{2})$.
In the light of the above statements, choose the correct answer from the options given below:

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x) - f(y)| \le 2|x - y|^{\frac{3}{2}}$ for all $x, y \in R$. If $f(0) = 1$,then $\int_{0}^{1} f^2(x) dx$ is equal to

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

Let $g(x) = x \cdot f(x)$,where $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$. Discuss the differentiability of $g$ at $x = 0$.