Harmonic(?) integral function

Question

This is the problem I'm trying to solve. It is in the Harmonic Function/Poisson Integral section of Papa Rudin. I tried directly computing the limit but I'm stuck. Any suggestions/ideas on how to get started would be great!

Answer 1

I will take $\phi$ in $L^{1}$ and show that the limit is $\phi (x)$ whenever x is a Lebesgue point of $\phi$. A simple calculation shows that $f(x+i\epsilon)-f(x-i\epsilon) = (g_\epsilon *\phi )(x)$ where $g_\epsilon (x) = \frac {2\epsilon} {x^{2} +\epsilon ^{2}}$. The functions $g_\epsilon$ form an approximate identity: they are non-negative, integrate to 1 (on the whole line; I am defining $\phi$ to be 0 outside the given interval) and $g_\epsilon (x) \to 0$ uniformly as $\epsilon \to 0$ provided $x$ stays away from 0. Hence $(g_\epsilon *\phi )(x) \to \phi (x)$ whenever x is a Lebesgue point of $\phi$. I will leave the other cases for you to complete.