Basic Analysis I — Real Analysis · Problem 5.5.10

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · improper-integrals

Problem

Find an example of an unbounded continuous function

f : [0, \infty) \to \mathbb{R}

that is nonnegative and such that

\int_0^{\infty} f < \infty

. Note that

\lim_{x \to \infty} f(x)

will not exist; compare previous exercise. Hint: On each interval

[k, k+1]

k \in \mathbb{N}

, define a function whose integral over this interval is less than say

2^{-k}