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I'm trying to show the following statement:

Let $f \in \mathcal{R}_{a} ^{b}$ be nonnegative and be continuous at some $x_0$ in the bounded, closed $[a,b]$. Also, let $f(x_0) \neq 0$. Show that $\int_a ^b f>0$.

I think that I should approach the problem by letting $L= \int_a ^b f = \lim_{||\mathcal{P}|| \rightarrow 0} S(f, \mathcal{P}, \eta)$ and showing that $L > 0$. Note that $\mathcal{P}$ is a partition of $[a,b]$ and $S$ is the Riemann sum of f determined by $\mathcal{P}$ and $\eta$.

(1) I'm not sure how to incorporate the requirement that $f$ be continuous at some $x_0 \in [a,b]$. Similarly, I can see that if $f(x_0) \neq 0$, then $f(a) \neq 0$ and $f(b) \neq 0$, but I'm not sure how to use that either.

(2) Would someone be able to verify whether my stated approach is correct? I would appreciate any guidance in getting started.

kathystehl
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    Continuity is absolutely necessary. If there's a "jump" at a point, the integral still might be $0$. –  Jun 18 '15 at 19:54
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    Have you learned about upper and lower sums? A good approach to such a problem as this is to construct a partition such that the lower sum L(f,P) is positive. (This is sufficient because you are given the function is integrable; the integral must be at least as big as any lower sum.) – Jordan Green Jun 18 '15 at 19:54
  • I have, and that is what I was planning on doing by when I said that I was going to show that $L>0$ (I was going to show it by establishing that the lower sum is positive, given a partition). This is where I currently am in my understanding: one needs to show both (i) that we have continuity at some point within $[a,b]$ and (ii) that, given a partition, the lower sum is positive (thereby showing that all the Riemann sums defined by this partition are positive). – kathystehl Jun 19 '15 at 13:05
  • However, I don't understand why we need to show that there is continuity at some fixed point, $x_0$ within $[a,b]$. Shouldn't f be continuous throughout $[a, b]$ since we are given $f \in \mathcal{R}_a ^b$? – kathystehl Jun 19 '15 at 13:09
  • My understanding is that if $f$ is Riemann integrable on $[a, b]$, then the function must have infinitely points of continuity on the interval. That is correct, yes? – kathystehl Jun 19 '15 at 13:14
  • @kathystehl. I suggest that you go back to Riemann integration definition. A Riemann integrable function needs not to be continuous. – mathcounterexamples.net Jun 19 '15 at 13:49
  • @mathcounterexamples.net I think that I am confused. I saw that in my book, and then I also saw this posting: http://math.stackexchange.com/questions/776398/a-riemann-integrable-function-must-have-infinitely-many-points-of-continuity?rq=1 – kathystehl Jun 19 '15 at 13:53
  • @mathcounterexamples.net Could you explain the difference between what you're saying and what is stated in the other posting that I've linked to? – kathystehl Jun 23 '15 at 15:23

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Hint.

If $f(x_0)>0$ then $f$ is stricly positive on a "small" interval of strictly positive length containing $x_0$.

mathcounterexamples.net
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  • So, we know that $\int_c ^d f(x)$ is positive for $(c,d) \subset [a,b]$ that form a neighborhood around $x_0$. Do you complete the proof considering $\int_a ^c f$ and $\int_d ^b f$? If you can show that these are positive since f is nonnegative on $[a,b]$, then shouldn't you have been able to directly show that these are positive without knowing that $f(x_0) \neq 0$? – kathystehl Jun 19 '15 at 13:02
  • I stated in my above comment my confusion about the continuity of $f$ in $[a,b]$. Perhaps, addressing that would clarify how one should treat the subsets of $[a,b]$ in which we don't necessarily have continuity. – kathystehl Jun 19 '15 at 13:12