The MGF is defined as $\mathbb E[e^{tX}]$ - in case this is finite for $t$ in a neighborhood of $0$, we say it exists.
Can there be a situation where the MGF exists but is not continuous (in $t$)? Specifically since we differentiate it and then evaluate it at $0$, can there ever be a situation where it is discontinuous at $0$?
I assume not, but the question is how to prove this.
