Let $f$ be continuously differentiable on $[0,1]$.
This gives us that $f$ is a semimartingale. I would like to understand why this is.
The definition of a semimartingale is a process that can be decomposed as a local martingale $M$ and an adapted cadlag process with bounded variation $A$.
So we should have $f(t) = M(t) + A(t), t \in [0,1]$.
We will have that that $f$ is of bounded variation. Is the local martingale $M(t)$ just such that it is trivially $0$ for all $t$? That is, $f(t) = M(t) + A(t) = 0 + A(t)$