Uniform convergence of piecewise continuous functions

Demon117

I like thinking of practical examples of things that I learn in my analysis course. I have been thinking about functions fn:[0,1] --->R. What is an example of a sequence of piecewise functions fn, that converge uniformly to a function f, which is not piecewise continuous?

I've thought of letting each of these sequences of functions being piecewise in terms of the fourier series but I am unsure this really works because I haven't figured out what it would converge to.

Any suggestions or other examples?