integration of function equals zero

Let $f$ be a continues function in $[a,b]$

$\forall x \in [a,b] \ \ \ f(x)\geq 0$

$ \int_{a}^{b}f(x)dx \ = 0$

Proof that $ \forall x \in [a,b] \ \ f(x) = 0 $

So how do I do that ?

What I know is:

because $f$ is continues function in $[a,b]$ I know it bounded and because $\forall x \in [a,b] \ \ \ f(x)\geq 0$ I know that $ \exists M \in \mathbb{N} \ \forall x \in [a,b] \ \ \ M \geq f(x)\geq 0$.

I also know that because $f$ is continues function in $[a,b]$ that $f \in R[a,b]$ meaning that there exists $F$ so that $F(a) = F(b)$, but how can I show that $ \forall x \in [a,b] \ \ F(b) = F(a) = F(x)$ ?

So how do I continue from here ?

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