If the integral of a continuous function is positive does it implies that the function is positive

Question

Let $f\in C[a,b]$. If we have for all $[c,d]\subset[a,b]$ $$\int_c^d f(x)dx\geq 0,$$ can we deduce that $f\geq 0 \; \forall \; t\in[a,b]$ ?

Answer 1

Yes, because then the map$$\begin{array}{rccc}F\colon&[a,b]&\longrightarrow&\mathbb R\\&x&\mapsto&\displaystyle\int_a^xf(t)\,\mathrm dt\end{array}$$is increasing and therefore $(\forall x\in[a,b]):F'(x)\geqslant0$. But $F'=f$.

Answer 2

Suppose no, then there is $\;t_0\in [a,b]\;$ for which $\;f(t_0)<0\;$ By continuity, there's an open neighborhood $\;I_0\;$ of $\;t_0\;$ (which can be one-sided if $\;t_0=a,b\;$) in which $\;f(t)<0\;$ . Deduce now a straighforward contradiction.