A visual proof that neural nets can compute any function

http://neuralnetworksanddeeplearning.com/chap4.html

One of the most striking facts about neural networks is that they can compute any function at all. That is, suppose someone hands you some complicated, wiggly function, f(x):
No matter what the function, there is guaranteed to be a neural network so that for every possible input, x, the value f(x) (or some close approximation) is output from the network, e.g.:
This result holds even if the function has many inputs, f=f(x1,…,xm), and many outputs. For instance, here's a network computing a function with m=3 inputs and n=2 outputs: