The universal approximation theorem states that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact subsets of Rn, under mild assumptions on the activation function.
How many neurons is needed? For a function in 3 variables and one output. And how many adjustable parameters does that give in the neural network?
There is not a particular number. You have to decide how much you want to approximate your function.
UAT, in informal language, means that neural network can approximate any given function f when given appropriate parameters (which are to be found experimentally; their learnability isn't a question here!).
If you have a function of 3 variables, then a neural network with 20 neurons may approximate it far better than one with the 10 neurons. It is possible that after a certain number of neurons, the difference between the outputs, i.e., (F - f) becomes very less.
I answer this question following Occam's razor, thus not mentioning any un-necessary terms. Please ask, if you have any additional or specific questions in it.
