# Thread: help por favor, uniform continuity

1. ## help por favor, uniform continuity

I have the function f [0, infinity) that is continuous on all R. I need to prove that f is uniformly continuous on [0, infinity) if either of the following are satisfied:
a.) there exists an a>0 such that f is uniformly continuous on [a, inf).
b.) the lim (as x approaches infinity) of f(x) exists as a real number.
Thank you for any suggestions.

2. Being able to help you is dependent on what you know.
Do you know that if a function f is continuous on a compact set then f is uniformly continuous there? Thus if $\displaystyle a>0$ then interval $\displaystyle \left[ {0,a} \right]$ is compact (closed and bounded) so f is uniformly continuous there. Also we are given that f is uniformly continuous on $\displaystyle [a,\infty )$, but $\displaystyle [0,\infty ) = \left[ {0,a} \right] \cup [a,\infty ).$

The part (b) is just a bit more difficult.
Say that $\displaystyle \lim _{x \to \infty } f(x) = r$ and $\displaystyle \varepsilon > 0$.
$\displaystyle \left[ {\exists N \in Z^ + } \right]\left( {x > N \Rightarrow \left| {f(x) - r} \right| < \frac{\varepsilon }{3}} \right).$
From above you can see that f is uniformly continuous on $\displaystyle \left[ {0,N} \right].$
Now prove that f is uniformly continuous on $\displaystyle [N,\infty )$.