uniform convergence?

i know the function f_n:[-1,1]-->reals where f_n(x)=x^(1/(2*n-1)) converges
pointwisesly to -1 (for -1</= x <0) and 1 for x>/= 0
what i want to know is when should a cts function converge to a limit which
is cts, is my thinking right when i say.
the reason this does not converge to a cts function (every though it is a
family of cts fctns) is due to the fact that the derivative tends to
infinity (f '_n(x)).
my question is ..does uniform convergence => derivatives converge? if so
could someone point out a theorem? (web search didnt come up with anything
helpful)
if this is not the case, is there a counter example (i dont believe this
case actually exists though!)

Many thanks


.

Loading