uniform continuity - Using Lipshitz always

Question

Question, is it possible to prove each question regarding uniform continuity with Lipshitz? I am having problem with this subject and can only answer be using Lipshitz. but I see there are some problems i can't manage to answer.
Let's say I must prove a function is uniform continuity on $(1,\infty)$ or like $(0,1)$ and such...
I never understand the meaning of the range while doing uniform continuity... just answering it using the Lipshitz (without thinking basically).
A month ago, I had a test - went really bad. anyway, there was a question:
prove $f(x)=x-cos^3\sqrt(x$) is uniform continuity at: $[0,\infty)$. I tried answering using Lipshitz, but I needed to prove first the derivative is blocked at $[1,\infty)$.
Anyway, you understand what i mean? Using Lipshitz, can I answer every question using Lipshitz? or I have to use Epsilion,delta \ other idea to solve?
Sorry for bad english.

Answer 1

If you have a problem in which you are supposed to prove that a certain function $f$ is uniformly continuous and if $f$ is not Lipshitz then, clearly, you cannot use the fact that the function is Lipshitz, since it is not. An example would be$$\begin{array}{rccc}f\colon&[0,\infty)&\longrightarrow&\Bbb R\\&x&\mapsto&\sqrt x.\end{array}$$