What the author means by "close approximation" is clarified later (under the "Two caveats" header). The point is that you can get an arbitrarily close approximation of any continuous function, just by choosing a sufficiently large number of hidden units. That is, for any epsilon > 0, a neural network can approximate any function within epsilon of the function's exact value at all points, with enough hidden units. The smaller the epsilon, the better the approximation you want, and the more hidden units you need.