Random variable $X$ with values depending on the coin flipping

Question

Let $X$ be random variable. Flip a coin, if the coin comes up head then $X=0$, if the the coin comes up tail then uniformly choose number in the interval $\left(0,1\right)$, let denote it $a$ and $X=a$. The task is to find distribution function $F(x)$ and how the density function $f(x)$ could look if we allow "infinite values". Any help will be appreciated. Thank you.

Answer 1

For $x < 0$ you have $P(X\le x)=0$ since $X$ does not take negative values. For $0\le x\le 1$ note that $$ P(X\le x) = P(\text{$X=0$ or $0<X\le x$}) = P(X=0) + P(X\in(0,x]). $$ Can you take it from there?