Cauchy coupling proves continuity of the half-plane Poisson extension ($\varphi (x+iy)=\mathbb{E}f(x+yC),C\sim \mathrm{Cauchy}(0,1)$)

For bounded continuous boundary data $f$, the half-plane Poisson kernel is the law of $x+yC$. On one common probability space, $x+yC\to x_0$ almost surely as $(x,y)\to (x_0,0)$ with $y>0$. Boundedness of $f$ and dominated convergence theorem imply $\varphi (x+iy)\to f(x_0)$. This proves continuity at every finite boundary point even for tangential approach. No moment of the Cauchy distribution is needed.