$f:\mathbb{R} \rightarrow \mathbb{R}$ is such that f'(x) exists $\forall x.$
And f'(-x)=-f'(x)
I would like to show $f(-x)=f(x)$
In other words a function with odd derivative is even.
If I could apply the fundamental theorem of calculus
\int_{-x}^{x}f'(t)dt = f(x)-f(-x) but since the integrand is odd we have $f(x)-f(-x)=0 \Rightarrow f(x)=f(-x)$
but unfortunately I don't know that f' is integrable.