As we know, the integral of an odd (integrable) function over a symmetric interval disappears. Now, I am having difficulties about proving a somewhat converse statement of this, specifically:
Let $f$ be a continuous, $2\pi$-periodic function on $\mathbb{R}$ such that $\int_{-\pi}^{\pi} f(x)\,e^{\,ikx} \; dx \; = \; 0$ for all even $k \in \mathbb{Z}$. Prove that f is an odd function.
I am not even sure if this statement holds. Maybe there is a counterexample which I don't see right now.