What are the 'iff' conditions for the integral of a periodic function over one period to be zero?

Question

Very simple question. Assume we are given a continuously differentiable $T$-periodic function $f: \Bbb R \to \Bbb R$, $f(x)=f(x+T)$. What are the necessary and sufficient conditions for the integral \begin{equation*} I = \int_0^Tf(x) dx \end{equation*}
to be identically zero? Clearly $f$ being odd is sufficient, but what are the necessary conditions if $f \neq 0$?

Answer 1

$f$ being odd is not necessary, as pointed out by TravisJ in the comments.

A simple sufficient condition for the integral $\begin{equation*} \int_0^Tf(x) dx \end{equation*}$ to be zero is : $f$ is the derivative of a $\mathcal{C}^1$ T-periodic function.

Let $g$ be a $\mathcal{C}^1$ periodic function such that $f=g'$, then :

$$\int_0^Tf(x) dx= \int_0^Tg'(x) dx=g(T)-g(0)=0.$$

Since you want $f$ to be $\mathcal{C}^1(\mathbb{R})$, then a necessary and sufficient condition is $f$ to be the derivative of a $\mathcal{C}^2$ $T$-periodic function.

Suppose $f$ a $\mathcal{C}^1(\mathbb{R})$ $T$-periodic function with : $\int_0^Tf(x) dx=0.$ Then $g(t)=\int_0^t f(x) dx$ exists, is $\mathcal{C}^2(\mathbb{R})$ and $T$-periodic.