The article demonstrates the theoretical foundations of the mathematical apparatus − the continuum of linear programming. It demonstrates a technique for solving problems with the use of orthogonal systems of functions. The article was an exact solution of the problem of variational calculus to linear constraints. The purpose of the work is to develop accurate methods of solving the problem in the class of Legendre polynomials. The study demonstrates an ability to build the exact solution of the problem and the conditions under which the decision is allowed. Based on the properties of Legendre polynomials, an exact solution of the problem of continual linear programming is provided, in which the integrands and functional limitations are presented in rows of finite degree. Analytically, it is proven that the solution obtained is a limiting case of the linear combination of delta functions. It is shown that the parameters of the optimization problem of finding the unknown functions plan contains half of the variables than in the canonical method. Recommendations are given for the construction of the optimization algorithm. There is a possibility of extending the proposed technology solution in the direction of using other systems of orthogonal polynomials.