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Let $f,g\colon [a,b]\rightarrow\mathbb{R}$ be continuous functions with $$\int_a^b f(t)dt= \int_a^b g(t)dt.$$ Then prove that there exists $c\in (a,b)$ such that $f(c)=g(c)$.

One way to do this is to consider $h(x)=\int_a^x (f(t)-g(t))dt$. Then by Fundamental Theorem of Calculus, $h$ is differentiable and $h(a)=h(b)=0$, and Rolle's theorem completes the solution.

I tried to proceed in the other way: suppose $f(c)\neq g(c)$ for any $c\in (a,b)$. Then $f-g$ is a continuous function on $(a,b)$ which is nowhere vanishing. Now, if $f(c)-g(c)>0$, then continuity guarantees that there is an open neighbourhood of $c$ in $(a,b)$ where $f-g$ is positive, and integration of a positive function is positive. Can we proceed further from here to arrive at a contradiction?

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    $\begingroup$ Unless I'm missing something, it seems like you have pretty much finished. Recall that $h(b) = 0$ and note that you just proved that $h(b)>0.$ $\endgroup$ – ktoi Jul 12 '15 at 13:52
  • $\begingroup$ Oh! its seems to be correct. $\endgroup$ – Groups Jul 12 '15 at 13:53
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By contradiction if $f(x) < g(x)$ for all $x \in [a,b]$ then $$\int_a^b f(x)dx < \int_a^b g(x) dx$$ as the integral of a stricly positive continuous function is strictly positive.

Similar proof if $g(x) < f(x)$ for all $x \in [a,b]$.

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  • $\begingroup$ you are considering ...for all $x\in [a,b]$. But both cases may arise? $\endgroup$ – Groups Jul 12 '15 at 13:52
  • $\begingroup$ A continuous function defined on an interval that takes positive and negative values vanishes. $\endgroup$ – mathcounterexamples.net Jul 12 '15 at 13:54
  • $\begingroup$ OK. Now it is clear. $\endgroup$ – Groups Jul 12 '15 at 13:55

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