It is written that the function $u$ is harmonic in $\mathbb{D}$ and continuous in $\overline{\mathbb{D}}$ and $u(e^{it})=\phi(t).$ I am unable to prove the continuity. z The continuity in the inside can be shown because it is harmonic and hence continuous (or since it is the convolution of two continuous functions). I am unable to show that it is continuous on $\overline{\mathbb{D}}$ and same as $\phi$ on the circle.