I am learning about moment generating function and characteristic function. Let $X$ be a real random variable. Let $M(t)=E[e^{tX}]$. Let $B=\{t\in\mathbb{R}:M(t)<\infty\}$. There’s a theorem that says $M$ is infinitely differentiable on $B^{\mathrm{o}}$, the interior of $B$. In particular, if we have $0\in B^{\mathrm{o}}$, then $M^{(k)}(0)=E[X^k]$.
I am wondering what would happen if $B=(-\infty,0]$. In this case, what would the one sided derivative of $M$ at $0$ mean?