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}^{\prime}\left(a,b\right) $ is the distributional limit of the analytic function $F$ defined in a region of the form $\left(a,b\right)\times\left( 0,R\right),$ if the one sided distributional limit exists, $f\left(x_{0}+0\right)=\gamma,$ and if $f$ is distributionally bounded at $x=x_{0},$ then the Lojasiewicz point value exists, $f\left(x_{0}\right)=\gamma$ distributionally, and in particular $F(z)\to\gamma$ as $z\to x_{0}$ in a non-tangential fashion