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$f:\mathbb{R} \rightarrow \mathbb{R}$ 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.

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