# Continuous but not uniformly continuous in $\text{GL}(2,\Bbb{C})$

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 MÃ¶bius 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)? – t.b. May 3 '12 at 2:55
Well, what's a function on $\mathbf R$ that isn't uniformly continuous? Let's start small. – Dylan Moreland May 3 '12 at 3:12
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... – t.b. May 3 '12 at 3:33
You must be confusing some things and misremember... You can't get much more continuous than being a polynomial in the entries. – t.b. May 3 '12 at 3:40
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. – t.b. May 3 '12 at 3:44