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The Itô formula decomposes the smooth transform into a continuous local martingale and a finite-variation process:If $f(M)$ has finite variation, then $N$ also has finite variation. A continuous finite-variation local martingale is constant, so $N=0$ andIt remains to show that the second-order term vanishes; the preceding observation alone does not establish this.

Let $Z=\{a:f^{\prime }(a)=0\}$. By the occupation-times formula for the local time of a semimartingale,Therefore $L_t^a(M)=0$ for almost every $a\notin Z$. Also $f^{\prime \prime }(a)=0$ for almost every $a\in Z$: if $f^{\prime }(a)=0$ and $f^{\prime \prime }(a)\ne 0$, continuity of $f^{\prime \prime }$ makes $f^{\prime }$ strictly monotone in a neighborhood of $a$, so that zero is isolated. Such isolated zeros form a countable set. Localizing to bounded paths and finite quadratic variation, then using the occupation-times formula once more, givesBoth terms in the Itô formula now vanish. Apply this on all rational times and use continuity to obtain the simultaneous conclusionThis is the finite-variation smooth transform of a continuous local martingale property.