If a nonnegative continuous function is positive at a point, its integral is positive

Question

Suppose that $f(x)\ge 0$ for all $x$ in $[a,b]$ and $f $ is continuous at $x_0 \in [a,b]$ and $f(x_0) > 0$. Prove that the integral from $a$ to $b$ of $f$ is greater than zero.

Can I prove this using the bounded theorem for integrals? Any suggestions on how to get started.

Answer 1

Since $f$ is continuos at $x_0$, there exists $\delta$ such that for any $x$ with $|x-x_0|<\delta$ it holds $f(x)>f(x_0)/2$. Thus $$\int_{x_0-\delta}^{x_0+\delta} f dx \geq f(x_0)\delta>0.$$ Use $f\geq 0$ to conclude.