I'm trying to show the following statement:
Let $f \in \mathcal{R}_{a} ^{b}$ be nonnegative and be continuous at some $x_0$ in the bounded, closed $[a,b]$. Also, let $f(x_0) \neq 0$. Show that $\int_a ^b f>0$.
I think that I should approach the problem by letting $L= \int_a ^b f = \lim_{||\mathcal{P}|| \rightarrow 0} S(f, \mathcal{P}, \eta)$ and showing that $L > 0$. Note that $\mathcal{P}$ is a partition of $[a,b]$ and $S$ is the Riemann sum of f determined by $\mathcal{P}$ and $\eta$.
(1) I'm not sure how to incorporate the requirement that $f$ be continuous at some $x_0 \in [a,b]$. Similarly, I can see that if $f(x_0) \neq 0$, then $f(a) \neq 0$ and $f(b) \neq 0$, but I'm not sure how to use that either.
(2) Would someone be able to verify whether my stated approach is correct? I would appreciate any guidance in getting started.
Hint.
If $f(x_0)>0$ then $f$ is stricly positive on a "small" interval of strictly positive length containing $x_0$.