examday

/TMUA

For this question we assume our given function $f(x)$ is differentiable and thus is (Riemann) integrable.

We say $f(x)$ is odd if and only if $f(-x)=-f(x)$.

Which of the following conditions are necessary for $f(x)$ to be an odd function?

I. $f(x)$ is a constant function.

II. $f(a-x)+f(a+x)=f(a)$, where $a$ is any real constant.

III. $\int_{-a}^a f(x)\,\dd x=0$, where $a$ is some real constant.

IV. $f'(x)>0$.