I need to show f(x)=x^(1/3) is uniformly continuous on (-1,1).
I can't find a delta st |x-y|<delta => |f(x)-f(y)|<epsilon..
any ideas?
You can try and find a delta but I am quite sure it's going to be hard, (been there done that), two aproaches that are easier are:
1) Show that it is a Lipschitz function (I'm not saying that it is, I don't know, haven't tried it)
2) Use extension theorem: a function is uniform continuous in (a,b) if and only if it can be defined in the end points a, b such that the extended function is continuous.
** Clearly the case of this function. The theorem is in the book Introduction to Real Analysis by Bartle.
It isn't Lipschitz since if is differentiable and Lipschitz then is bounded. which is unbounded as
This only works for the exact reason that Plato said namely a continuous function on a compact metric space (in this case simply ) is automatically u.c.2) Use extension theorem: a function is uniform continuous in (a,b) if and only if it can be defined in the end points a, b such that the extended function is continuous.
** Clearly the case of this function. The theorem is in the book Introduction to Real Analysis by Bartle.