Can a Continuous and Integrable Function Have an Infinite Limit?

[lynxman72]

Hi all, I'm looking for a positive real-valued function definition on all of R such that the function f(x) is continuous and integrable (the improper integral from -infinity to infinity exists and is finite) but that lim sup f(x)=infinity as x goes to infinity. I'm thinking about something with spikes of decreasing width but I'm not sure how to get an exact formula for the function or if this is the right idea. Any help is much appreciated.

 

[AKG]

Yes, spikes (triangles) of growing height and shrinking width is the right idea. Let your function be 0 for negative x. On the right, look at intervals [n, n+1) for n = 0,1,2,... Make these intervals [n,n+1) correspond to a number a_n such that the series:

$$\sum _{n=0}^{\infty}a_n < \infty$$