wonderings

I think that provided there is some open interval (x_n-δ,x_n+δ) in which every point had been mapped to the continuous and differentiable function after uncountably many steps, then the function at x_n would be continuous and differentiable.
The problem is that any function f_n(x) does not fit this criterion, because you are going one point at a time, so it's a countably infinite number, so we can do a diagonal proof to find a point however close to x_n that is still 0, and so at no time would it be continuous.