I am doing problems in Real Analysis by Stein, and was wondering if my solution to the following question is valid:
I had to find a Lebesgue integrable continuous function whose limit superior diverges as the modulus of $x$ tends to infinity. My choice is as follows: Divide the whole of $\mathbb{R}$ into closed intervals of the form $\left[n,n+1\right]$. Place a triangle of base 1 and height 1 on the interval $\left[0,1\right]$ and $\left[-1,0\right]$. On $\left[-2,-1\right]$ and $\left[2,1\right]$, place the triangles of base length $1/4$ and height $2$. In general, in every interval, place a triangle whose base is a quarter of its previous(if the interval is on the positive portion of $\mathbb{R}$) one, and whose height is twice that of the previous one. Define the function to be zero where there is no triangle. Clearly this function is continuous everywhere, and its integral converges. However, the limit superior of this function, as the modulus of $x$ tends to infinity, is also infinite.
Does it look okay?
I can't find any flaw; I think it's ok.