1) Let's assume that the integral of f(x) between a and infinity is convergent and that f(x) is equally continuous in [a, +infinity). Prove that the limit of f(x) when x tends to infinity is zero.
Okay so I assume negatively that the limit of f(x) when x tends to infinity is either different from zero or doesn't exist at all. How do I continue?
2) Let f(x) be a positive and integrable function in [0,t] to every t>0. Let's decide a>0 and define h(t) to be the integral of f(x) between t-a and t+a. t>=a. Prove that if the integral of f(x) between zero and infinity is convergent, then the integral of h(t) between a and infinity is convergent.
Have no idea how to solve this...not even a direction
For example, suppose that f(x) is zero except for a narrow spike around each integer value of x. More precisely, suppose that f(x) goes linearly from 0 to 1 in the interval , and then goes down linearly from 1 to 0 in the interval The area under this triangular spike is , and the total area under the curve is finite because converges. But f(x) does not tend to 0 because it keeps jumping up to 1 and back again.