The universal approximation theorem guarantees that a finite-width neural network that approximates the function to within some epsilon exists. But, regardless of the approximation method, there is no way to certify that a given approximation method is sufficient for an arbitrary continuous function given only a finite number of samples (i.e., without oracle knowledge of the underlying function), which is the typical situation where neural networks are applied. I can construct a continuous function that has an arbitrary (but non-infinite) number of peaks in an arbitrary interval. Thus, any method that approximates the function within some epsilon for all possible inputs within that arbitrary interval must encode an arbitrary amount of information. I can also ensure that whatever the number of samples is, it's not enough to properly approximate the function.