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Math Help - If f is Riemann integrable and continuous, prove that F is differentiable

  1. #1
    Member Last_Singularity's Avatar
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    If f is Riemann integrable and continuous, prove that F is differentiable

    Some help on this following question, please. Thanks a bunch!

    Question: If f is Riemann integrable on [a,b] and continuous at x_0, prove that F(x) = \int_{a}^{x} f(t) dt is differentiable at x_0 and F'(x_0)=f(x_0). Show that if f has a jump discontinuity at x_0, then F is not differentiable at x_0.
    Last edited by Last_Singularity; April 22nd 2009 at 02:56 PM.
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  2. #2
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    I sure that you mean F(x) = \int_a^x {f(t)dt} .

    Notice that F(x_0  + h) - F(x_0 ) = \int_{x_0 }^{x_0  + h} {f(t)dt} and the length of that interval is |h|.
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  3. #3
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    Consider the difference quotient:  \frac{F(x)-F(x_0)}{x-x_0} = \frac{\int_{a}^{x} f(t) \ dt- \int_{a}^{x_0} f(t) \ dt}{x-x_0} = \frac{1}{x-x_0} \int_{x_0}^{x} f(t) \ dt .

    Then you need to show that  \lim\limits_{x \to x_0} \frac{1}{x-x_0} \int_{x_0}^{x} f(t) \ dt = f(x_0) .

    Use the fact that  f(x_0) = \frac{1}{x-x_0} \int_{x_0}^{x} f(x_0) \ dt . Also use the fact that  \left| \int_{a}^{b} f \right| \leq \int_{a}^{b} |f| provided that  f is Riemann integrable. You have to consider two cases:  x_0 < a and  x_0 > a .
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