# 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.

• What's the purpose of the subscript $0$ ? – Yves Daoust Oct 28 '18 at 13:44
• Are you asking about the existence of a continuous function $\phi$ such that $\phi(\cos x)=\sin x$ over $[0,2\pi)$ ? – Yves Daoust Oct 28 '18 at 13:45
• There is no purpose for the subscript 0. Yes. That's my question, @YvesDaoust – asd11 Oct 28 '18 at 13:51

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