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

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

$F(x)=\{\begin{array}{ccc}0& \text{f}\text{o}\text{r}& x<mn\; id="EditorElement\_1782906">0</mn>\\ x& \text{f}\text{o}\text{r}& 0x<mfrac\; id="EditorElement\_1782943"><mrow\; id="EditorElement\_1782936"><mn\; id="EditorElement\_1782933">1</mn></mrow><mrow\; id="EditorElement\_1782942"><mn\; id="EditorElement\_1782939">4</mn></mrow></mfrac>\\ \frac{1}{4}& \text{f}\text{o}\text{r}& \frac{1}{4}x<mfrac\; id="EditorElement\_1783002"><mrow\; id="EditorElement\_1782995"><mn\; id="EditorElement\_1782992">1</mn></mrow><mrow\; id="EditorElement\_1783001"><mn\; id="EditorElement\_1782998">2</mn></mrow></mfrac>\\ x^2& \text{f}\text{o}\text{r}& \frac{1}{2}x<mn\; id="EditorElement\_1783046">1</mn>\\ 1& \text{f}\text{o}\text{r}& 1x\end{array}$

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

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

b$.$ The $25c.$ The median for $P$ is uniquely determined

d$.$ The distribution function for $P$ is differentiable over the entire $R.$