= Finite-variation smooth transform of a continuous local martingale
{title2=$f(M_t)=f(M_0)$}
If $M$ is a <continuous local martingale> and $f\in C^2(\mathbb R)$, then <finite variation> of $f(M)$ forces $f(M_t)=f(M_0)$. The <Itô formula> and the <quadratic variation of a stochastic integral> give $\int_0^t f'(M_s)^2d[M]_s=0$. The <occupation-times formula> implies that $L_t^a(M)=0$ almost everywhere where $f'(a)\ne0$. On the zero set of $f'$, its derivative $f''$ vanishes almost everywhere: zeros at which $f''\ne0$ are isolated and hence countable. Thus $\int_0^t|f''(M_s)|d[M]_s=\int |f''(a)|L_t^a(M)da=0$. Both terms in the <Itô formula> vanish.
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