Suppose that we start with some continuous function $f$ defined on $[a,b]$. Since it is continuous it is integrable so the number $\int\_{a}^{b}f(x)dx$ exists.

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> How to show (in an as elementary as possible way) that there exist continuous function $g$ such that we have $f \neq g$ and $\int\_{a}^{b}(f(x)-g(x))dx=0$?
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This is geometrically obvious but how to prove it?