  In this article, we study an analog of the Bj\"orling problem for isothermic
surfaces (that are more general than minimal surfaces): given a real analytic
curve $\gamma$ in ${\mathbb R}^3$, and two analytic non-vanishing orthogonal
vector fields $v$ and $w$ along $\gamma$, find an isothermic surface that is
tangent to $\gamma$ and that has $v$ and $w$ as principal directions of
curvature. We prove that solutions to that problem can be obtained by
constructing a family of discrete isothermic surfaces (in the sense of Bobenko
and Pinkall) from data that is sampled along $\gamma$, and passing to the limit
of vanishing mesh size. The proof relies on a rephrasing of the
Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its
discretization which is induced from the geometry of discrete isothermic
surfaces. The discrete-to-continuous limit is carried out for the Christoffel
and the Darboux transformations as well.
