Prove that the integral of a function from a to b is greater than zero given the following criteria

Question

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.

Answer 1

(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.