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I did my research on this question and I found two answers that I feel contradict one another: this first one says it's not true and this second one says it is. When I asked my professor about it he said the following:

Let $f : (a,b) \rightarrow \mathbb{R}$ be a uniformly continuous function. Then $f$ can be continuously extended to $[a,b]$

Then if $f : [a,b] \rightarrow \mathbb{R}$ is a continuous function, then it is also uniformly continuous.

Therefore if $f : (a,b) \rightarrow \mathbb{R}$ is a uniformly continuous function, this implies $f : [a,b] \rightarrow \mathbb{R}$ is also uniformly continuous.

I'm inclined to believe that the second and my professor are correct, but the example of the first one is pretty compelling.

Thomas Grenier
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  • You've got a good (+1) answer, but for posterity: If $f$ is uniformly continuous on $(a, b)$, (1) that does not mean $f$ is defined at all on $[a, b]$; (2) if $f$ is defined on $[a, b]$ that does not mean $f$ is continuous on $[a, b]$; (3) strictly speaking it's abuse of notation to call both functions $f$. A function comes equipped with a domain and a codomain. Changing the domain, for example, technically changes the function. – Andrew D. Hwang Nov 23 '23 at 16:10

1 Answers1

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$f$ can be extended to a uniformly continuous function on $[a,b]$ but not every extension is uniformly continuous. In fact, not every extension is even continuous, as the example in that link shows.

There is a unique continuous extension and this extension is uniformly continuous.

geetha290krm
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