Continuous functions and Neural Networks Is there any continuous function $\phi$ such that $\phi(\cos x) = \sin x$ over $0,2\pi)$? If so, could you give me an example? I stumbled across this problem after trying to train a single layer neural network to do the same thing as my purported continuous function. By the [Universal Approximation Theorem I figured that if I can't train the neural net (training error is very high) it means there isn't a continuous function there to approximate the neural net towards. NN are dense in the space of continuous functions. Any thoughts? EDIT: If it's relevant, I trained a radial basis function neural net.

There cannot be such a function because $\phi(\cos(2\pi-x))=\phi(\cos x)$ while $\sin(2\pi-x)=-\sin x$.