Here is the statement which has been given :
Let $A \in \mathbb{R}^*_+$, $f \in \mathcal{C}^1([0,A],[0,f(A)])$ a strictly increasing function with $f(0)=0$. Then for all $x \in [0,A]$ and $y\in [0,f(A)]$, $\displaystyle xy \le \int_{0}^{x}f(u)\mathrm{d}u \ + \int_{0}^{y}f^{-1}(u)\mathrm{d}u$.
In the proof given, the $\mathcal{C}^1$ aspect is used to justify that the function $\displaystyle \phi : [0,A] \to \mathbb{R},\ y \mapsto xy- \int_{0}^{x}f(u)\mathrm{d}u \ - \int_{0}^{y}f^{-1}(u)\mathrm{d}u$ has maximum at $f(x)$ and $\phi(f(x))=0$. On the other, the homeomorphism's theorem in one variable, the sign of the derivative and the fundamental theorem of the integral calculus have been used.
I was just wondering if we could weaken the hypothesis on $f$ to generalize the statement ($\mathcal{D}$ for differentiable on $[0,A]$ seems to work for instance) ?
Thanks in advance !
