I understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. For example, my book defines continuity as:
Definition 4.3.1. A function $f:A \to \mathbb R$ is continuous at a point $c \in A$ if, for all $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x-c| < \delta$ (and $x \in A$) it follows that $|f(x)-f(c)| < \epsilon$.
Uniform continuity is defined as:
Definition 4.4.5. A function $f:A \to \mathbb R$ is uniformly continuous on $A$ if for every $\epsilon > 0$ there exists a $\delta > 0$ such that $|x-y| < \delta$ implies $|f(x)-f(y)| < \epsilon$.
I know that in Definition 4.3.1, $\delta$ can depend on $c$, while in definition 4.4.5, $\delta$ cannot depend on $x$ or $y$, but how is this apparent from the definition? From what appears to me, it just seems like the only difference between Definition 4.3.1 and Definition 4.4.5 is that the letter $c$ was changed to a $y$.
My guess is that the first definition treats $c$ as a fixed point and it is only $x$ that varies, so in this case, $\delta$ can depend on $c$ since $c$ doesn't change. Whereas for the second definition, neither $x$ or $y$ are fixed, rather they can take on values across the whole domain, $A$. In this case, if we set a $\delta$ such that it depended on $y$, then when we pick a different $y$, the same $\delta$ may not work anymore. Is this somewhat a correct interpretation?
Anymore clarifications, examples, would be appreciated.
Best Answer
First of all, continuity is defined at a point $c$, whereas uniform continuity is defined on a set $A$. That makes a big difference. But your interpretation is rather correct: the point $c$ is part of the data, and is kept fixed as, for instance, $f$ itself. Roughly speaking, uniform continuity requires the existence of a single $\delta>0$ that works for the whole set $A$, and not near the single point $c$.