A number of assumptions seem to be missing from this article. Since the author is using the sigmoid function which is smooth, this argument actually only applie

A number of assumptions seem to be missing from this article. Since the author is using the sigmoid function which is smooth, this argument actually only applies to approximating smooth functions. That is, you dont just need f to be continuous, you need all of its derivatives to exist and all be continuous. Also, since we are only able to have finitely many neurons, we need to be able to approximate f using step functions with definitely many pieces. So this argument can only be used if f is constant outside of a compact region.