I'm having a bit of trouble understanding uniform continuity in "plain English" (for the lack of a better term).
The definition of continuity I'm working with is this:
Let $f: A\ \subseteq \mathbb{R} \rightarrow \mathbb{R}$.
We say that $f$ is continuous at $x_o \in A$ if, for every $\epsilon > 0$, there is some $\delta > 0$ s.t. if $|x – y| < \delta$, then $|f(x) – f(x_0)| < \epsilon$.
If I'm understanding this correctly, then continuity at $x_0$ means that we can draw any "box" of dimensions $\delta$ x $\epsilon$ such that every conceivable $f(x)$ will lie within that box (and this box can be arbitrarily small).
$f$ is continuous over $(a, b)$ if the above definition holds for all values within that interval.
But, the definition of uniform continuity is confusing to me. The definition I'm working with is this:
Let $f: A \subseteq \mathbb{R} \rightarrow \mathbb{R}$.
We say that $f$ is uniformly continuous if for all $\epsilon > 0$, there is some $\delta = \delta (\epsilon) > 0$ s.t. if $x, y \in A$ with $|x – y| < \delta$, then $|f(x) – f(y)| < \epsilon$.
I'm having a lot of trouble understanding what this definition is saying. First, why are we setting $\delta$ to $\delta * \epsilon$? Secondly, if possible, how do we put the above definition in terms of the "box" analogy (or some better analogy)?
In what ways is uniform continuity different from continuity? Any help understanding unif. continuity and the difference between these 2 definitions would be greatly helpful. Thank you.
Best Answer
The following is a (hopefully) intuitive explanation. With regular continuity, the following happens: I give you a point $(a,f(a))$ and an epsilon. Your job is to find a delta such that if $x$ is at most delta away from $a$, then $f(x)$ is at most epsilon away from $f(a)$. Continuity says that we can find these deltas for a given point and epsilon.
Uniform continuity steps things up a bit. Now, Let's say I take some set $A$, we'll say its a subset of the real line. Uniform continuity gives us an epsilon first, and picks an $\epsilon$ interval/ball inside $f(A)$, and doesn't tell us what point it is centered at. Uniform continuity then tells us that we have some $\delta(\epsilon) $ ($\delta$ depends on $\epsilon$ )such that if $|x-y | <\delta$, then necessarily $|f(x)- f(y)|<\epsilon$. In other words, for a uniformly continuous function, given any epsilon region of our codomain, we can find a delta region of our domain such that the choice of any two points $x,y$in this $\delta$ region gives us the property that $|f(x)-f(y)|<\epsilon$. In other words, we have control over how "close together" $f$ can be on the whole set, rather than just around specific points.