Characterize all the continuous functions that satisfy the following condition

Question

Characterize all the continuous functions $f$ in $[a,b]$ that satisfy $$\int_a^b f(x)\,\varphi(x)\,dx = 0$$ $\forall\,\varphi(x)$ continuous in $[a,b]$ such that $\int_a^b \varphi(x)\,dx = 0$. I understood that the condition is verified by constant function, but I don't know how to prove it. I tried for reductio ad absurdam, but I have no idea how to do it.

Answer 1

Hint:

If $\Phi$ is the set of all functions $\phi(x)$ whose integral over $[a,b]$ is null, then in particular $$ \sin \left( {\omega \left( {x - {{a + b} \over 2}} \right)} \right)\quad \left| {\;\forall \omega \ne 0} \right. $$ belong to $\Phi$.

Fourier series …

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Consider the functions $\phi \in \Phi$ which are continuous in $[a,b]$ and odd wrt the midpoint of the segment.
Then $f(x)$ shall be even over $[a,(a+b)/2,b]$.

Now consider the functions which are continuous over all $[a,b]$, odd wrt a sub-segment therein, and null outside: then $f(x)$ shall be even over such a sub-segment.

We conclude that $f(x)$ must be even over all $[a,b]$ and any sub-segment of it, i.e. $f(x)=const$.