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 $f\in\mathcal{D}^{\prime}(a,b)$ 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, $f(x_{0}+0) =γ,$ and if $f$ is distributionally bounded at $x=x_{0}$, then the Łojasiewicz point value exists, $f(x_{0})=γ$ distributionally, and in particular $F(z)\to γ$ as $z\to x_{0}$ in a non-tangential fashion.

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