Question

Let f : R → R and g : R → R be continuous functions. Then. the value of the integral

Answer 1

Let f:R→R be a continuous function such that ∫∞0f(x)dx exists. Then which of the following statements are correct?

(a) If limx→∞f(x) exists, then limx→∞f(x)=0,

(b) The limit limx→∞f(x) must exist and is zero,

(c) In case f is a nonnegative function, the limit limx→∞f(x) must exist and is zero,

(d) In case f is a differentiable function, the limit limx→∞f′(x) must exist and is zero.

Answer 2

Let's say limx→∞=alimx→∞=a. if a≠0a≠0, then by definition of limit, there is a number XX such that for any x≥Xx≥X, we have |f(x)|≥a/2|f(x)|≥a/2. This means that from that point on, the integral ∫x0f(x)dx∫0xf(x)dx will increase (or decrease, depending on whether aa is positive or negative) by at least a/2a/2 for each unit we increase xx, and thus the improper integral doesn't exist. So if aa exists and the integral is finite, then a=0a=0 by this contradiction.
Statement (b) and (c):
I do these together because they can be disproven with the same counter-example. Start out with the function that is constantly 00. Now, take the interval [0.5,1.5][0.5,1.5], and raise the graph as a triangle with a top in the point (1,2)(1,2). Now the total integral is 11. Then for the interval [1.75,2.25][1.75,2.25], raise a triangle so the top is at (2,2)(2,2). The total integral is now 1.51.5. Continue doing this around every integer point along the xxaxis, always halving the interval length. You will end up with a function that integrates to 22, but has bumps all over with height