  If $\ph$ is an analytic function bounded by 1 on the bidisk $\D^2$ and
$\tau\in\tb$ is a point at which $\ph$ has an angular gradient
$\nabla\ph(\tau)$ then $\nabla\ph(\la) \to \nabla\ph(\tau)$ as $\la\to\tau$
nontangentially in $\D^2$. This is an analog for the bidisk of a classical
theorem of Carath\'eodory for the disk.
  For $\ph$ as above, if $\tau\in\tb$ is such that the $\liminf$ of
$(1-|\ph(\la)|)/(1-\|\la\|)$ as $\la\to\tau$ is finite then the directional
derivative $D_{-\de}\ph(\tau)$ exists for all appropriate directions
$\de\in\C^2$. Moreover, one can associate with $\ph$ and $\tau$ an analytic
function $h$ in the Pick class such that the value of the directional
derivative can be expressed in terms of $h$.
