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, $f(x_{ 0}+0) =\gamma,$ 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.
Categories: math.CV math.FA