A continuous function that is uniformly continuous on two sets, but not uniformly continuous on the union of these two sets?

Question

This homework problem has just cost me 3 hours... But I still have no clue what it can be...

Let $A, B \subseteq \mathbb{R}$. Find a continuous function $f:A\cup B \to \mathbb{R}$ where $f$ is uniformly continuous on $A$ and on $B$, but $f$ is not uniformly continuous on $A\cup B$.

The wording "Let $A,B\subseteq\mathbb R$" seems to imply that we're supposed to do this for arbitrarily given sets $A$ and $B$. I'm sure that wasn't intended; we get to pick the sets as well as the function. Badly worded problem. — bof, Oct 31 '13 at 06:50
The question is wrong if $A,B$ are compact subset of $\mathbb{R}$ — Myshkin, Oct 31 '13 at 05:51
There is no restriction on what $A$ and $B$ could be as long as they are subsets of the reals. — Qingtian, Oct 31 '13 at 05:55

detnvvp · Answer 1 · 2013-10-31T05:39:31.007

3

Hint: take $A=\mathbb N$ and $B=\{n+\frac{1}{n}\left|\right.n\in\mathbb N,n\geq 2\}$.

edited Oct 31 '13 at 05:39

answered Oct 31 '13 at 05:32

detnvvp

8,237

$f$ is meant to be continuous on $A \cup B$. – Oct 31 '13 at 05:34
Oh, right; I edited the hint. – detnvvp Oct 31 '13 at 05:39
@detnvvp Why $A \cup B$ is not discrete? – GA316 Jan 22 '14 at 17:53
@GA316 But, it is discrete. – detnvvp Jan 22 '14 at 20:08
@detnvvp so $f$ will be uniform in $A \cup B$ also no, since any function is uniformly continuous on discrete set? – GA316 Jan 23 '14 at 06:48
@GA316 Any function defined on a discrete set is continuous, but not necessarily uniformly continuous. – detnvvp Jan 23 '14 at 08:54
@detnvvp ya. ya. now I understand. very nice. Thanks. – GA316 Jan 23 '14 at 18:57

score 1 · Accepted Answer · answered Oct 31 '13 at 06:55

1

$$f(x)=\cos x^2, A=f^{-1}(1),B=f^{-1}(-1)$$

answered Oct 31 '13 at 06:55

bof

6,156

score 1 · Answer 3 · answered Nov 09 '13 at 10:52

1

Hint : union of two discrete sets need not be discrete

answered Nov 09 '13 at 10:52

GA316

4,324
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A continuous function that is uniformly continuous on two sets, but not uniformly continuous on the union of these two sets?

3 Answers3