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Math Help - show these statements of riemann intergrable

  1. #1
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    show these statements of riemann intergrable

    Show these statements are true for continuous function f, but false for Riemann integrable functions f.

    1. If f:[a,b] --> R is such that f(t)>=0 for all t in [a,b] and
    integral( f(t), a, b) =0 then f(t)=0 for all t in [a,b]

    2. integral(f(x), a, t) is differentiable and the derivative = f(t)
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  2. #2
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    Quote Originally Posted by szpengchao View Post
    1. If f:[a,b] --> R is such that f(t)>=0 for all t in [a,b] and
    integral( f(t), a, b) =0 then f(t)=0 for all t in [a,b]
    Use the continuity property, if f(x_0) > 0 as some point x_0 \in [a,b] then f(x) > 0 on [a,b]\cap (x_0-\delta , x_0+\delta) for some \delta > 0. And then the integral cannot possibly equal to 0. When it is not continous just take a single point jump as a conterexample.

    2. integral(f(x), a, t) is differentiable and the derivative = f(t)
    The continous case follows from the fundamental theorem of calculus. Again use the same conterexample involving a point jump. The derivative of the integral at the point just is not equal to the original function at that point.
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