Let $f$ be continuous on $[a,b]$ and suppose that $f(x)\ge0$ for all $x \in\ [a,b].$
Prove that if there exists a point $c \in\ [a,b]$ such that $f(c)>0$, then $\int_{a}^b f(x)\,dx > 0 .$
I feel like by proving this function is uniformly continuous, which is trivial, that it somehow shows me that the integral is greater than $0$, but I don't quite know how to get there. Can someone help?
Thanks! On a side note, I don't think the mathtex stuff came out right, let me know how to fix that!