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 $c\in[a,b]$?


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*The same question assuming Lebesgue integrability.
Thanks


As another example, take $f(x)=\frac{1}{\sqrt{x}}$ for $x \in [\epsilon, 1]$ and $f(x)=\frac{1}{\sqrt{\epsilon}}$ for $x\in[0,\epsilon]$ (with fixed $\epsilon$ of course).
Then $f$ is nonnegative, continuous and its integral is less than $\int_0^1 \frac{\mathrm{d}x}{\sqrt{x}}=2$. But $f$ can take values as large as $\frac{1}{\sqrt{\epsilon}}$, which is larger than $2$ for small enough $\epsilon$.
