A simple function with no tangential limits but with non-tangential limits

Question

I am going to be teaching a course about the Hardy space and I would like to show the students that the non-tangential limit is a necessary concept BEFORE telling them about Blaschke products. Is there any simple way to construct a function in the Hardy space for which the non-tangential limits exist but the tangential limits fail everywhere (or a.e.) to exist?

Answer 1

Let $f$ be a function on the unit disk $D$; let $f^*$ denote its boundary values (defined via nontangential limits, a.e.).

If the unrestricted limit $\lim_{z\to\zeta,\ z\in D } f(z)$ exists at a boundary point $\zeta$, then $f^*$ has a limit at $\zeta$. Indeed, otherwise we would have a sequence $\zeta_n\to\zeta$ such that $\zeta_n\to\zeta$ but $ f(\zeta_n)$ does not converge. Pick $z_n\in D$ such that $|z_n-\zeta_n|<1/n$ and $|f(z_n)-f(\zeta_n)|<1/n$. It follows that $f(z_n)$ does not converge.

So, an example you want is provided by any Hardy space function with nowhere continuous boundary values. For example, $$f(z)=\sum_n c_n(1-z/\zeta_n)^{-\epsilon}$$ where $\{\zeta_n\}$ is dense in the unit circle, and $\epsilon, c_n$ are small enough so that the sum converges in $H^p$.