Suppose $f(x) > 0 $ is a pdf that is strictly positive on the entire real line. Must $f$ be continuous?
I've thought about trying to say that the corresponding cdf must be positive on any interval and trying to use that, but I'm not even sure if that is true.
Can someone provide me with any insight?
NO. Consider $f(x)=\frac 1 4$ for $|x| \leq 1$ and $f(x)=\frac {3/4} {x^{4}}$ for $|x| \geq 1$.