Value of a function is than its integral. Suppose that $f(x)$ is nonnegative and Riemann integrable on $[a,b]$, is it true that 
$ f(c)\leq \int_{a}^{b}f(x)dx$
for any $c\in[a,b]$?


*

*The same question assuming Lebesgue integrability.
Thanks


No, take $f(x)=x$ on $[0,1]$:
$$
\int_0^1 x\,dx=\frac{1}{2}
$$
while $f(1)=1$.
This of course is a counterexample also for the second question.
Both for Lebesgue and Riemann integrals, in any case, you can't deduce nothing about the value at a point, because the integral doesn't change if you modify the value on a finite set of points.
