In this paper, we propose a method for approximating the solution of the linear Fredholm integral equation of the second kind which is defined on a closed contour Г in the complex plane. The right-hand side of the equation is a piecewise continuous function that is numerically defined on a finite set of points on Г. To approximate the solution, we use a linear combination of B-spline functions and Heaviside step functions defined on Г. We discuss both theoretical and practical aspects of the pointwise convergence of the method, including its performance in the vicinity of the points where discontinuities occur.