Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$

I take it for granted that you know that this integral is never negative. So you need to give a single interval over which it's positive.

$f(x_0) > 0, f(x)$ continuous there means that there is a $\delta$ such that for $|x-x_0|< \delta$, $|f(x) - f(x_0)| < \frac{f(x_0)}{2}$.

So consider the interval $[x-\delta, x+\delta]$.

I think this is the easiest way to do it, perhaps.

I think this problem can be solved with Reductio ad absurdum. If it is not true, then the integral must equal to zero, so the upper sums must equal zero. This means the supremum in each interval is equal to zero, therefore $f(x) = 0$ in every subinterval, so $f(x) = 0$ for all $x$. This contradicts the fact that $f(x_0) > 0$ for some $x_0$.