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Math Help - Integral Inequalities

  1. #1
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    Integral Inequalities

    Suppose  f and  g are real valued functions that are Riemann integrable on  [a,b] . Suppose further that  f(x) \leq g(x) for all  x \in [a,b] and that a strict inequality holds for at least one point in  [a,b] . Prove that  \smallint_{a}^{b} f < \smallint_{a}^{b} g need not hold. If we add a continuity condition to  f and  g , does the above inequality hold?

    Proof. We know that  \smallint_{a}^{b} f \leq \smallint_{a}^{b} g . Define  f and  g as follows:
     f(x) = \begin{cases} 0 \ \ \ \ \text{if} \ x \neq c \\ K \ \ \  \text{if} \ x = c \end{cases}

     g(x) = \begin{cases} 0 \ \ \ \ \text{if} \ x \neq c \\ L \ \ \ \ \text{if} \ x = c \end{cases}

    Here we have  c \in [a,b] and  K < L . Then  \smallint_{a}^{b} f = \smallint_{a}^{b} g = 0 .  \diamond


    If we had a continuity condition, then I think the above inequality will hold? Do we use the following fact: There exists an open interval  (c,d) in  [a,b] such that  x \in (c,d) and  f(u)>0 for all  u \in (c,d) ? Is this correct?
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  2. #2
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    Quote Originally Posted by manjohn12 View Post
    If we had a continuity condition, then I think the above inequality will hold? Do we use the following fact: There exists an open interval  (c,d) in  [a,b] such that  x \in (c,d) and  f(u)>0 for all  u \in (c,d) ? Is this correct?
    You have the correct idea what contunity. This is similar to the proof of this result: \int_a^b f = 0 \implies f = 0 \text{ if }f\geq 0 \text{ and continous}.

    Basically what you do is assume that f(x_0) > 0 and then you know in an interval of this point we have f>0 which leads to a contradiction. You can apply this similar argument to f-g.
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