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QUESTION

(a) Suppose $f:[a,b] \to \mathbb{R} $ is nonnegative, continuous on $[a,b]$, and not identically zero. Prove that:

$\int_{a} ^{b}f(x)\, dx > 0$

(b) If we replace in part (a) the assumption of continuity by the assumption of integrability on $[a,b]$, show that the conclusion of part (a) is not true.

Paramanand Singh
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Kelly
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  • You should mention the third (and slightly difficult to prove) alternative : if $f$ is positive and Riemann integrable on $[a, b] $ then the integral is positive. See this answer https://math.stackexchange.com/a/519921/72031 – Paramanand Singh May 07 '18 at 03:21

1 Answers1

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(a) Not identically 0 implies there exists c, $a\lt c\lt b$, where f(c)> 0. Continuity implies there is a non-zero length interval (h) around c, where $f(x)\gt \frac{f(c)}{2}$. Therefore the integral > $\frac {hf(c)}{2}$> 0.

(b) Let f(x)= 0 on the interval, except on a finite number of points and let f(x)=1 on those points. The integral is still 0.

herb steinberg
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