Let (left(B_{t}ight)_{t geqslant 0}) be a (mathrm{BM}^{d}, f: mathbb{R}^{d} ightarrow mathbb{R}) be a continuous function such that

Question:

Let $\left(B_{t}ight)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}, f: \mathbb{R}^{d} ightarrow \mathbb{R}$ be a continuous function such that $\int_{0}^{t} f\left(B_{s}ight) d s=0$ for all $t>0$. Show that $f\left(B_{s}ight)=0$ for all $s>0$, and conclude that $f \equiv 0$.

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