Re: Uniform continuity


Jthompson5...@xxxxxxxxx wrote:
We've been going over uniform continuity in my analysis class, and have
shown several examples of functions bring uniformly continuous, such as
f(x) = 6x+3 on [0, +infinity), and g(x) = x^4 on [-1,2], but I was just
wondering for myself if the function h(x) = x^a is uniformly continuous
on [0, +infinity) for any integer a. In the case that a>1, it seems as
if we have only looked at closed, bounded intervals. Is this just a
coincidence, or is there a reason for this? Does anyone have any kind
of proof? Thanks,

JT

There is a theorem which states that any continuous function on a
closed, bounded set is uniformly continuous (note: in the real numbers,
the closed, bounded sets are precisely the compact sets).
On [0,+infinity), x ^ a will not be uniformly continuous for a > 1.

.

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