Let $f$ be a continuous and integrable function on $[a,b]$ such that
$$\int_a^b f(x)\,\mathrm{d}x = 2$$
and for every $t_1,t_2$ such that $\displaystyle t_2 -t_1 = \frac{b-a}2$
$$\int_{t1}^{t2} f(x)\,\mathrm{d}x =1$$
Prove that $f$ is periodic and then determine its period.
The previous subquestion asked to prove that if $f$ is continuous then there exists an $F$ such that $F'(x)=f(x)$ for every $x$ and it's supposed to help.
Frankly I have no idea where to even start from.
I want to prove that there exists a $T$ such that $f (x+T) = f (x)$ for every $x$ that's all I can say, any insights please?
