Answer the following questions a$,$ b$,$ c$,$ d with true or false

Let $P$ be the probability measure on $R$ that is given by the distribution function

$F(x)=\{\begin{array}{ccc}\frac{1}{2}{e}^x& \text{f}\text{o}\text{r}& x<mn\; id="EditorElement\_2324874">0</mn>\\ 1-\frac{1}{2}{e}^{-x}& \text{f}\text{o}\text{r}& x0\end{array}$

Determine for each of the following statements whether it is true or false.

a$.F$ is differentiable on the whole $R.$

b$.P$ has a density with respect to the Lebesgue measure.

c$.P$ has a point probability of $\frac{1}{2}$ at $0.$

d$.P$ has a uniquely determined quantile function.