Sequence of differentiable functions, non-differentiable limit
I am trying to find a sequence of differentiable functions which converge uniformly on [-1,1] but such that the uniform limit is NOT differentiable on (-1,1).
I figure I'd like to make something converge to f(x)=|x| (which is not differentiable at 0, so not differentiable on (-1,1) ). I have an idea, but haven't been able to hack through the details. The idea is to define each member of the sequence such that outside of [-1,1] each function is identically |x|, but on (-1,1) I want to mutate x^2 somehow so that the sequence converges uniformly to |x|...and them's the details I haven't hacked through.