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.
Thank you very much! That is so much nicer than my attempt. I'm going to apply that sequence for sure, but still I am curious to know if my attempt could be made to work.
Hopefully I'll have time to give it some mind and if I do I'll put up my efforts (but I'm a few weeks behind in coursework so for now I'll just move on).
Here is another possiblity. For all n, for x< 0, if , f(x)= 1 for x> 1/n, is continuous for all x (but NOT differentiable at x= 1/n). The sequence converges to f(x)= 0 for x< 0, 1 for which is not continuous at x= 0.
So integrate that: for x< 0, for , for x> 1/n is differentiable for all n (in particular, its derivative at 0 is 0, and at 1/n is 1) but its limit, F(x)= 0 for x< 0, x for which is not differentiable at x= 0.