Providing a counterexample: If f is cont. on [0,infty), f unbounded, then..."

[bb8]

Proving, or providing a counterexample:

"If f is a continuous function on [0, ∞) and f is unbounded, then the improper integral 0 to ∞ f(x)dx has to diverge."

I know that this statement is false, but how can I go about providing a counterexample. I know that f(x) would have to be piecewise in order for it to work, but I can't think of a counterexample. Thank you!!

 

[stapel]

Proving, or providing a counterexample:

"If f is a continuous function on [0, ∞) and f is unbounded, then the improper integral 0 to ∞ f(x)dx has to diverge."

I know that this statement is false, but how can I go about providing a counterexample.

How do you "know" that this is false? What example do you have in your head, which proves this to you? (This example may be a good counter-example.)

I know that f(x) would have to be piecewise in order for it to work...

"In order for" what "to work"? The counter-example? The integral? The function? How do you "know" this?

Please be complete. Thank you!

 

[pka]

"If f is a continuous function on [0, ∞) and f is unbounded, then the improper integral 0 to ∞ f(x)dx has to diverge."

I will gladly provide you with a general outline. However, I will not work out details because they are just too darn messy.

At each \(\displaystyle n\in\mathbb{Z}^+ \) map \(\displaystyle n\mapsto n \). Then construct a tent(isosceles triangle) of height \(\displaystyle n\) and length of base \(\displaystyle \left(n\cdot 2^{n}\right)^{-1} \). Outside the 'tents' the function is zero.

Please respond showing your efforts.