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Math Help - riemann integrable

  1. #1
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    riemann integrable

    Suppose that f is continuous on [a,b], that f(x)≥0 for all x within [a,b] and that . Prove that f(x)=0 for all x within [a,b]. Also, show that the continuity hypothesis cannot be dropped.
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  2. #2
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    Here is a hint.
    If \left( {\exists c \in [a,b]} \right)\left[ {f(c) > 0} \right] then \left( {\exists [d,e] \subseteq [a,b]} \right)\left( {\forall z \in [d,e]} \right)\left[ {f(z) > \frac{{f(c)}}{2}} \right].
    Ask yourself what is the minimum value of \int_d^e {f(x)dx} .

    Now drop continuity: f(x) = \left\{ {\begin{array}{rl}<br />
   {1/n} & {x = a + 1/n}  \\<br />
   0 & {else}  \\<br /> <br />
 \end{array} } \right. .
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  3. #3
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    Quote Originally Posted by Plato View Post
    Here is a hint.
    If \left( {\exists c \in [a,b]} \right)\left[ {f(c) > 0} \right] then \left( {\exists [d,e] \subseteq [a,b]} \right)\left( {\forall z \in [d,e]} \right)\left[ {f(z) > \frac{{f(c)}}{2}} \right].
    Ask yourself what is the minimum value of \int_d^e {f(x)dx} .

    Now drop continuity: f(x) = \left\{ {\begin{array}{rl}<br />
   {1/n} & {x = a + 1/n}  \\<br />
   0 & {else}  \\<br /> <br />
 \end{array} } \right. .
    Tired messing around with your hint but could not get anywehre.
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