Can a Continuous and Integrable Function Have an Infinite Limit?

lynxman72
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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.
 
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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 an such that the series:

\sum _{n=0}^{\infty}a_n < \infty
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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