Let X be a random variable. Define the function $g(x) = \frac{P(X < x)+P(X \leq x)}{2} \forall x$ Which of the following statements is always true?
(A) g is right continuous at all $x \in \mathbb{R}$.
(B) g is not a monotone function.
(C) g is left continuous at all $x \in \mathbb{R}.$
(D) $\lim_{x \to \infty}g(x)=1$
We know that the CDF $F(x)=P(X \leq x)$ is monotone, right continuous and its limit is 1 as $x \to \infty$. So $g=F(x) - \frac{P(X=x)}{2}$. If X is a continuous random variable then $P(X=x)=0$, so $g(x)=F(x)$ and (A),(D) seem to be correct, but only one option is true and I'm not sure what happens when X is discrete, please help.