So, to recognise multiple digits, does the network have multiple outputs??
Generally, and that would be easiest, but you don't _have_ to do it that way (for example you could do thresholding on a single output).
And if you had a big enough network, could you train it to recognise anything?
Thoeretically, but totally impractical. Humans have REALLY big intricately architected neural networks that have been fed for a lifetime with a nearly continuous stream of data using an incredibly advanced training algorithm and even we can't always recognise faces (if you just turn photos upside down it becomes difficult for us to even recognize photos of friends). @karligula
And once you HAVE trained the network, you could theoretically extract the function that it represents?
GroundKeepers reply is spot on, but I would also add a few comments. While our abilities to extract symbolic functions from even the simplest of neural networks is pretty bad, but can sometimes be done, there are a also a couple of problems with the very idea of converting networks to symbolic functions...
When we can represent a system with a function instead of network we generally go straight to using the function instead of the network. So, for simple things where a network or a fucntion would both do we usually already try to use the function.
For the types of problems where we do use neural networks for, the weights and architecture of the network is itself often the most compact notation for the function we can come up with because our symbolic notation just isn't well suited to represent the kinds of functions that neural networks are well suited for.