Let $f[a,b]: \mathbb R \rightarrow\mathbb R$ be a (Riemann) integrable function such that $f \geq0$ and $\int_{a}^{b}f = 0$. Verify with counterexamples that the sentence aforementioned conditions do not imply $f\equiv 0$.
I couldn't think of any functions except for the case when $a = b$, which is trivial. More than some counterexamples, I'd like to know how to be able to think on a number of them by myself, I mean, which I ideas should I hold on to in this particular exercise.
For example you can set: $$ f(x) = \begin{cases} 0 & x \not = c \\ K & x = c\end{cases}$$
with $c \in [a,b]$ and $0 < K < \infty$ and you get, for every possible values of $c$ and $K$: $$ \int_a^b f(x) \textrm{d}x= 0$$
The trick is that integral "doesn't see" set of null measure (points). Given a function $g_1(x)$, if you modify it only in some points defining $g_2(x)$ you will have: $$\int_a^b g_1(x)\textrm{d}x = \int_a^b g_2(x)\textrm{d}x$$
consider $f(a)= 8$ and $f(x)=0$ for every $x\neq a$, then $f$ satisfies the terms and obviously not always zero
