  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.
