  Computing the rate-distortion function for continuous sources is commonly
regarded as a standard continuous optimization problem. When numerically
addressing this problem, a typical approach involves discretizing the source
space and subsequently solving the associated discrete problem. However,
existing literature has predominantly concentrated on the convergence analysis
of solving discrete problems, usually neglecting the convergence relationship
between the original continuous optimization and its associated discrete
counterpart. This neglect is not rigorous, since the solution of a discrete
problem does not necessarily imply convergence to the solution of the original
continuous problem, especially for non-linear problems. To address this gap,
our study employs rigorous mathematical analysis, which constructs a series of
finite-dimensional spaces approximating the infinite-dimensional space of the
probability measure, establishing that solutions from discrete schemes converge
to those from the continuous problems.
