Integral over absolute values and null function

Question

Let $f:[a,b]\to\mathbb{R}$ be continous. How can I show that $$ \int_a^b |f(x)|dx=0\text{ iff }f(x)=0\text{ f.a. }x\in [a,b] $$

Answer 1

Assuming $f$ is continuous:

Let $F$ be a primitive of $|f|$ on $[a,b]$, Since $|f|\geqslant 0$, $F$ is increasing and $a<b$ implies that $F(a)\leqslant F(b)$. Thus, $\int_a^b |f(x)|dx \geqslant 0$.

If $\int_a^b |f(x)|dx = 0$ then $F(a)=F(b)$ and since $F$ is increasing, it is constant over $[a,b]$ therefore its derivative $|f|$ is zero.

The reciprocal is fairly obvious.