PCDF

Construct a cumulative distribution function $F(x)$ of a random variable which matches the probability density function of another random variable whenever $F(x)\neq 1$. How many different sorts can you make?

Could you make a cdf $G(x)$ which could be used as a pdf for all values of $x< \infty$ ? Give as clear a reason as possible.

Can you create an example in which the cumulative distribution function $F(x)$ of a random variable $X$ and the probability density function $f(x)$ of the same random variable $X$ are identical whenever $F(x)< 1$?