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Inspired by distribution of (inverse) distribution function. This made me think: is the following statement true?

Let $X$ be a random variable with support in $[a, b]$. $F_{X}$, the cumulative distribution function of $X$, is strictly increasing in $[a, b]$ if and only if $X$ is a continuous random variable.

By "strictly increasing," I mean that if $x, y \in [a, b]$, then $f(x) < f(y)$. I don't know enough measure theory to be able to prove this statement, but it seems like it would be true, as I can't think of a counterexample. I imagine this has something to do with the properties of the Lebesgue-Stiltjes integral, but again, I don't know enough to be able to tackle this problem.

[This is not a homework question.]

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Clarinetist
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No. Consider $[0,1]$ with the distribution function $$F(x) = \begin{cases} \frac{1}{4}x, & 0 \le x < \frac{1}{2} \\ \frac{1}{4}(x-1) + 1, & \frac{1}{2} \le x \le 1.\end{cases}$$ Then $F_X$ is strictly increasing and right continuous, $F(0)=0$ and $F(1)=1$, so it is a distribution function. But the random variable $X$ whose cdf is $F$ has $P(X = \frac{1}{2}) = \frac{1}{2}$ so $X$ is not continuous.

One way to construct such an $X$: Let $Z$ be a coin flip and let $U$ be a uniform random variable which is independent of $Z$. Set $X=U$ if $Z$ is heads, and $X=\frac{1}{2}$ if $Z$ is tails.

The correct necessary and sufficient condition for $X$ to be a continuous random variable is that $F_X$ be absolutely continuous.

Nate Eldredge
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    Fascinating. I've never heard of absolute continuity. Clearly I need to learn more about measure theory to understand some of the statements in that page.

    Is the statement $X \text{ is continuous }\Leftrightarrow F_{X} \text{ is absolutely continuous}$ a standard theorem proved in a probability theory text, and could you refer me to a source that has this statement?

     – Clarinetist Dec 06 '14 at 20:24
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    @Clarinetist: Yes, it's pretty standard. It basically follows from a generalization of the fundamental theorem of calculus. I didn't find it in the first couple probability texts I checked, but you can find a sufficient statement as Theorem 3.35 of Folland's Real Analysis. – Nate Eldredge Dec 06 '14 at 20:31