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## Homework Statement

If ##f:[0,1] \rightarrow \mathbb{R}## is a continuous function such that ##\int_{0}^{x} f(x) dx = \int_{x}^{1} f(x) dx##, ##\forall x \in [0,1]##, show that ##f(x) = 0## ##\forall x \in [0,1]##.

## Homework Equations

## The Attempt at a Solution

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I have tried doing this by assuming ##\exists c \in [0,1]## such that ##f(c) > 0##. however i don't think it is bearing any fruit. I thought maybe then i could make a partition of [0,c] and [c,1] and show that their upper sums dont equal, but I dont really know where to start to show this either.

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