In this paper we propose an efficient algorithm for approximating piecewise continuous functions, defined on a closed contour Γ in the complex plane. The
function, defined numerically on a finite set of points of Γ, is approximated by a
linear combination of B-spline functions and Heaviside step functions, defined on Γ.
Theoretical and practical aspects of the convergence of the algorithm are presented,
including the vicinity of the discontinuity points.
