  A complex manifold $Y$ is said to have the interpolation property if a
holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a
holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be
a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker
property called the convex interpolation property. By a deep theorem of
Forstneri\v{c}, the two properties are equivalent. They (and about a dozen
other nontrivially equivalent properties) define the class of Oka manifolds.
  This paper is the first attempt to develop Oka theory for singular targets.
The targets that we study are affine toric varieties, not necessarily normal.
We prove that every affine toric variety satisfies a weakening of the
interpolation property that is much stronger than the convex interpolation
property, but the full interpolation property fails for most affine toric
varieties, even for a source as simple as the product of two annuli embedded in
$\mathbb{C}^4$.
