When (for what kind of $f$ and e.g. $a \in \mathbb{R}$) is $$\int_0^{2a} f= 2 \int_0^a f$$?
With some trigonometric functions ($a=\pi$)?
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For all $a$? Very rarely. For some $a$? Quite frequently. – Feb 17 '16 at 17:28
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1Should hold for the cases when $f(x + a) = f(x)\ \forall x,\ a\in\mathbb{R}$, but there may well be others – jameselmore Feb 17 '16 at 17:28
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Assume $f$ continuous and differentiate with respect to $a$: $$ 2f(2a)=\frac{d}{da}\int_0^{2a}f=2\int_0^a f=2f(a). $$ Thus, $f(2a)=f(a)=f(2^{-1}a)=\cdots=f(2^{-n}a)\to f(0)$, due to continuity.
Thus $f(a)=f(0)$, for all $a$, and finally $f$ has to be constant..
Yiorgos S. Smyrlis
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