You missed this paragraph, which occurs in the first 10% of the article. The second caveat is that the class of functions which can be approximated in the way

You missed this paragraph, which occurs in the first 10% of the article. The second caveat is that the class of functions which can be approximated in the way described are the continuous functions. If a function is discontinuous, i.e., makes sudden, sharp jumps, then it won't in general be possible to approximate using a neural net.