Continuity Confusion

Question

I was reading from Introduction to Analysis by Bartle and I read the following theorem about uniform continuity titled "Uniform Continuity Theorem":

"Let $I$ be a closed bounded interval and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous on $I$. Then $f$ is uniformly continuous on $I$. "

I know that uniform continuity can be defined as being continuous at every point in the domain. My confusion is coming from "continuous ON $I$". To me, it sounds like when you say a function is "continuous on $I$" it is saying that it is continuous on the entire interval $I$ and therefore uniformly continuous on $I$. However, it seems that the wording of the "Uniform Continuity Theorem" suggests "continuous on" and "uniformly continuous" are different. Could somebody please clarify for me?

Answer 1

The definition for continuity of a function $f:X\to\mathbb R$ at a point $x$ where $X\subset\mathbb R$ is: $$\forall\epsilon>0\quad\exists\delta>0\quad\forall y\in X\quad |x-y|<\delta\Rightarrow|f(y)-f(x)|<\epsilon$$ And $f$ is continuous on a set $Y\subset X$ when: $$\forall x\in Y\quad\forall\epsilon>0\quad\exists\delta>0\quad\forall y\in X\quad |x-y|<\delta\Rightarrow|f(y)-f(x)|<\epsilon$$ The definition for uniform continuity on $Y$ is: $$\forall\epsilon>0\quad\exists\delta>0\quad\forall x\in Y \quad\forall y\in X\quad |x-y|<\delta\Rightarrow|f(y)-f(x)|<\epsilon$$

There is a subtle difference in the order of the quantifiers between these two. This is the difference between pointwise and uniform continuity and you should carefully study the definitions.

Edit: had my quantifiers in the wrong order

Answer 2

in closed bounded intervals , continuity at $I $ and uniform continuity at $I $ are equivalent. This is not true for other intervals. For example

$x\mapsto x^2$ is continuous at $[0,+\infty) $ but not uniformly continuous.

$x\mapsto \frac {1}{x} $ is continuous at $(0,1] $ but not UC.