Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Probability / 9 / a / ii / Solution

Every cumulative distribution function is nondecreasing, right-continuous, valued in $[0,1]$, and has limits $0$ at $-\infty$ and $1$ at $+\infty$. For a continuous random variable it is continuous. When a probability density function exists, it is in fact absolutely continuous, with nonnegative derivative almost everywhere.