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Essentially, I'm trying to find an example of a function that is continuous but not uniformly continuous on $\text{GL}(2,\Bbb{C})$. I'm aware that this group is isomorphic (up to constant multiples) to the Möbius transformations. So I'm having a go at developing some geometric intuition there. However, as my prior inquiries indicate, my epsilon-delta skills are lacking. Hence, I find these sorts of examples tough to fabricate.

Edit: Norm is the boring Euclidean one on $\Bbb{C}^4$.

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    What metric do you put on GL(2,C)?2012-05-03
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    @t.b. Apologies! The boring one. Euclidean Norm...viewing it as a vector in C^4...2012-05-03
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    So, how about devising a non-uniformly continuous function on $\mathbb{C}^4$ first, then?2012-05-03
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    @t.b. This was my approach...struggle still medoes...2012-05-03
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    Well, what's a function on $\mathbf R$ that isn't uniformly continuous? Let's start small.2012-05-03
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    What @Dylan said...2012-05-03
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    @DylanMoreland x^2 seems pretty cool.2012-05-03
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    @t.b. (didn't want you to miss out on the sweet action)2012-05-03
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    Next step: embed an unbounded part of $\mathbf{R}$ isometrically into $GL(2,\mathbf{C})$ and devise a function on $GL(2,\mathbf{C})$ that restricts to $x^2$ on that embedded copy.2012-05-03
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    @t.b. and I'm lost...2012-05-03
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    Try $A \mapsto (\det{A})^2$ and the embedding $t \mapsto \begin{pmatrix} t & 0 \\ 0 & 1\end{pmatrix}$, for $t \neq 0$, for example...2012-05-03
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    @t.b. Ah, much thanks! Apologies for my lack of mental fortitude. And one last question...I know that det is discontinuous over R. Can you briefly remind me why C fixes things?2012-05-03
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    You must be confusing some things and misremember... You can't get much more continuous than being a polynomial in the entries.2012-05-03
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    @t.b. oops. path-connectedness. that's enough failures for 1 night.2012-05-03
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    I suggest that you try to concoct a few more examples and post them as an answer. Then ping me and I'll have a look.2012-05-03

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