On distributional point values and boundary values of analytic functions

We give the following version of Fatou’s theorem for distributions that are boundary values of analytic functions. We prove that if $f\in {\mathcal{D}}^{\mathrm{\prime }}(a,b)$ is the distributional limit of the analytic function $F$ defined in a region of the form $(a,b)\times (0,R),$ if the one sided distributional limit exists, $f(x_0+0)=\gamma ,$ and if $f$ is distributionally bounded at $x=x_0$, then the \L ojasiewicz point value exists, $f(x_0)=\gamma$ distributionally, and in particular $F(z)\to \gamma$ as $z\to x_0$ in a non-tangential fashion.