1. ## continuous function, integral

Suppose that $\displaystyle f: [a,b] \rightarrow \mathbb{R}$ is continuous and $\displaystyle f(x) \geq 0$ for all $\displaystyle x \in [a,b]$. Prove that if $\displaystyle \int^b_a f(x)dx=0$, then $\displaystyle f(x)=0$ for all $\displaystyle x \in [a,b]$.
Attempt
I had attempted to do this problem by contradiction, except I did not understand how to finish the problem. I would appreciate a few helpful hints on this one.

2. ## Fundamental Proof

Originally Posted by selenne431
Suppose that $\displaystyle f: [a,b] \rightarrow \mathbb{R}$ is continuous and $\displaystyle f(x) \geq 0$ for all $\displaystyle x \in [a,b]$. Prove that if $\displaystyle \int^b_a f(x)dx=0$, then $\displaystyle f(x)=0$ for all $\displaystyle x \in [a,b]$.
From the proof in the fundamental theorem of calculus:

$\displaystyle F(b) - F(a) = \lim_{\| \Delta \| \to 0} \sum_{i=1}^n \,[f(c_i)(\Delta x_i)]\$

$\displaystyle F(b) - F(a) = \int_{a}^{b} f(x)\,dx\$

With $\displaystyle f(c_i)\geq 0$ from given $\displaystyle f(x) \geq 0$ for all $\displaystyle x \in [a,b]$

And the definite integral evaluated as $\displaystyle \int^b_a f(x)dx=0$ from given, then
$\displaystyle \lim_{\| \Delta \| \to 0} \sum_{i=1}^n \,[f(c_i)(\Delta x_i)]=0\$

with $\displaystyle f(c_i)$ confined to be non negative, the summation can only be zero if $\displaystyle f(x)=0$ for all $\displaystyle x \in [a,b]$.