Theorem

Let $\mathcal{E}_{b}$ ($b$ for bounded, $\mathcal{E}$ for simple predictable processes) be the space of processes of the form $$X=H_{0}\mathbb{1}_{(t_{0},t_{1}]}+H_{1}\mathbb{1}_{(t_{1},t_{2}]}+\dots+H_{k}\mathbb{1}_{(t_{k},t_{k+1}]}$$where $H_{i}$ is $\mathcal{F}_{t_{i}}$-measurable $\forall i$, and $X$ uniformly bounded. Define $\mathcal{I}:\mathcal{E}_{b}\to L^{0}(\Omega,\mathcal{F},P)$ and let $M=(M_{t})_{t\ge 0}$ s.t. $$\mathcal{I}(X)=H_{0}(M_{t_{1}}-M_{t_{0}})+\dots+H_{k}(M_{t_{k+1}}-M_{t_{k}})$$$\mathcal{I}$ is Continuous (for uniform convergence on $\mathcal{E}_{b}$ and convergence in probability in $C^{0}$) if and only if $M$ is a semimartingale. i.e. $$\mathcal{I}\text{ continuous}\iff M\text{ semimartingale}$$