Let $f: [a,b]\to\mathbb R$ integrable. The following statements are equivalent:
$1) \int_a^b |f(x)|dx=0;$
$2) \mbox{ if } f \mbox{ is continuous at point } c \mbox{ then } f(c)=0;$
$3) \mbox{ int}(X=\{x\in[a,b]; f(x)\neq 0\})=\emptyset.$
For $1\Longrightarrow 2$ i tried:
Since $f$ is integrable, it follows that $|f|$ is integrable and $$ \left|\int_a^b f(x)dx\right|\leq\int_a^b|f(x)|dx=0 \Longrightarrow \int_a^b f(x)dx=0. $$
Moreover, since $f$ is continuous in $c$, then the function $$ F(x)=\int_a^x f(t)dt $$ is derivable in $c$ and $F'(c)=f(c)$.
Now, if $F'(c)\neq0$ then either $F'(c)>0$ or $F'(c)<0$. Assume $F'(c)>0$. Thus, there is $\delta>0$ such that $$ x,y\in(a,b) \mbox{ and } c-\delta<x<c<y<c+\delta \Longrightarrow F(x)<F(c)<F(y). $$
I'm stuck here. Any tips?