Let $f: [0,2] \rightarrow \mathbb{R}_{+}^{\ast}$ be a continuous function such that $\int_ {0}^{1} f(t) dt = \int_ {1}^{2} f(t) dt = 1$. Prove that there is a function $g: [0,1] \rightarrow [1,2]$ of class $C^{1}$, such that $\int_{x}^{g(x)} f(t) dt=1$.
Can anybody help me? Can I conclude that the function is periodic? or symmetrical in the range $[0,1]$ and $[1,2]$. What can I get out of it?