Let $C \subset \mathbb{R}^m$ be a compact and convex set. Consider a function $f: C \rightarrow \mathbb{R}$ continuously differentiable on $C$ such that $$\int_C \, f(x) \, d\lambda(x) \, = \, 0 $$ where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}^m$.
Is it possible under this assumptions to show that $f \equiv 0$ on the boundary $\partial C$? Otherwise, under which conditions, it holds that?
Any help/suggestion would be appreciate. Thanks in advance!
No, you cannot show that $f$ vanishes on the boundary. Consider the example $m=1$, $C=[-1,1]$, $f(x)=x$.