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**nmatthies1** (a) Let $\displaystyle g:[a,b] \to \mathbb{R}$ be a continuous function. Suppose that $\displaystyle g(x) \geq 0 ~\forall x \in [a,b]$ and $\displaystyle \int_{a}^{b} g = 0$.

Prove that $\displaystyle g(x)=0 \forall x \in [a,b] $.

Intuitively, this statement is obviously true. However, I can't seem to find a starting point for a formal proof. Where to start? Any hints?

(b) Give an example of a regulated function $\displaystyle h:[a,b] \to \mathbb{R}$ such that $\displaystyle h(x) \geq 0 ~ \forall x \in [a,b]$, $\displaystyle \int_{a}^{b} h = 0$, and there is a point $\displaystyle c\in [a,b]$ such that $\displaystyle h(c) \neq 0$.

My problem here is that I don't really understand the difference between continuous and regulated functions. I do know the formal definition of a regulated function, and the fact that a continuous function is regulated but do not entirely grasp the concept.

Also, it seems rather counterintuitive to assert that there would be a point $\displaystyle c$ such that $\displaystyle h(c) \neq 0$.