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    Riemann integrable

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    Are these functions Reimann Integrable? I am just learning this topic, so my description may not be accurate. A function is Reimann Integrable if it's Upper Darboux Sums and Lower Darboux suns are equal.

    Or stated another way, if U(f, P) - L(f, P) < e

    The two functions are piecewise functions.

    1) f(x) = { 0 when x = 0
    1/n when x is in (1/(n+1) , 1/n]

    Show that F(x) is Reimann Integrable on [0,1]

    2) f(x) = { sin(PI/x) if 0 < x <= 1
    0 if x = 0

    Also show that F(x) is Reimann Integrable

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    Solution Summary

    This solution helps explore Reimann Integrable functions within the context of real analysis.