Uniform Continuity

uniform continuity

Let $f : A \to R$ is uniformly continuous on $A$ if $\forall ε > 0$ $\exists δ > 0$ such that $x, y \in A$ has:

| x - y | < δ \Rightarrow | f (x) - f (y) | < ε

In contrast, Continuity itself is continuous at a point $y$ specifically. Having to consider the $y$ may imply that $δ$ depends on that $y$ . Here, we don't care about specific points it is continuous at.