Approximation of singularity sets with analytic graphs over the ball in C$^{2}$

Let $h$ be a smooth function on the ball in C$^{2}$ whose gradient has length less than or equal to 1. We show that if $h$ is uniformly near an analytic function on every complex affine one-dimensional slice then it must be near some function analytic on the whole ball. We use this to show the following: a singularity set over the ball which is near the graph of a function $h$ with $|\nabla h|\leq 1$ must be near the graph of some analytic function over the ball.