Let $f(x)$ be a monotonic increasing function on $[0,\infty)$ and for every $x>0$ it is integrable in $[0,x]$, so that $\lim_{x\to \infty}\frac{1}{x}\int_{0}^{x}f(t)dt=a$. I need to prove that $\lim_{x\to \infty}f(x)=a$.
I tried to use the limit definition: $|\frac{1}{x}\int_{0}^{x}f(t)dt -a|<\epsilon$ and to use the fact that $\int_{0}^{x}f(t)dt\geq x\inf f(x)$, but I can't see how does it help me eventually.
I would love your help with this.
Thanks a lot!