In metric spaces, is a function uniformly continuous iff $\delta$ depends on $\varepsilon$?

Question

Most book examples end with an expression for $\delta$ that depends on $\varepsilon$ when proving uniform continuity. What I am wondering is whether a function can be uniformly continuous as long as the distance between any two elements of the domain of the metric space is within some number that does not depend on $\varepsilon$ or the where we "fix" the domain.

Answer 1

There exist bounded, non uniformly continuous functions, so the answer is no. One example of such a function is $f(x)=\cos(x^2)$, for $x\in\mathbb R$. Then, for all $x,y\in\mathbb R$, you have that $|\cos(x^2)-\cos(y^2)|\leq 2$, but $f$ is not uniformly continuous on $\mathbb R$.