Given a fixed level of precision for input and output, and a function on this discrete space, we can construct a continuous extension of this function to the re

Given a fixed level of precision for input and output, and a function on this discrete space, we can construct a continuous extension of this function to the reals. Now using the paper we know that there is a neural network with continuous weights that approximates this continuous extension to arbitrary precision. If we imagine rounding the continuous input and output to the fixed precision specified, then because of continuity of the neural network, and the fact there is a finite number of weights, we can choose a tolerance by which we can change each of the weights such that the output does not change by more than the precision of the output value. Thus we can pick a level of precision for the weights where both the weights and inputs and outputs are discrete.