I have read a notion from a number of different sources simply stating that a step function can approximate any trigonometric function. 

I am not convinced by simply reading this notion, for example the step function is discontinuous where the step is defined to be, while the trigonometric functions (sine and cosine) are continuous. How can the approximation happen? Does the fact that the step function only has finite discontinuities matter?

I cannot find any sufficient remarks or proofs on this notion anywhere online and I have searched for about two days till now. Any help would be greatly appreciated!