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Integration Proof Question?

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  • 1 year ago

    If f(x) < x for ALL x in [0,1], then the area would be less than 1/2, because the graph of f would lie entirely below the top edge of the triangle described by f(x) = x on [0,1].

    If f(x) > x for ALL x in [0,1], then the area would be greater than 1/2, same reasoning.

    If f(x) = x for all x in [0,1], there is nothing to prove, we are finished.

    The only remaining cases are cases where SOME x's in [0,1] have f(x) less than x, while some other x's in [0,1] have f(x) greater than x. But then the graph of (a continuous) f(x) must cross the line f(x) = x, somewhere.

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