Putting it all together with an example

As we already mentioned, multi-layer neural networks can classify linearly separable classes. In fact, the Universal Approximation Theorem states that any continuous functions on compact subsets of Rn can be approximated by a neural network with at least one hidden layer. The formal proof of such a theorem is too complex to be explained here, but we'll attempt to give an intuitive explanation using some basic mathematics. We'll implement a neural network that approximates the boxcar function, in the following diagram on the right, which is a simple type of step function. Since a series of step functions can approximate any continuous function on a compact subset of R, this will give us an idea of why ...

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