When doing calculus with respect to a non-Archimedian metric (such as the p-adic metric), is it true that differentiability does NOT imply continuity (example: the p-adic norm seems to be differentiable but not continuous)?

Printable View

- May 10th 2014, 01:39 PMSlipEternalNon-Archimedian calculus
When doing calculus with respect to a non-Archimedian metric (such as the p-adic metric), is it true that differentiability does NOT imply continuity (example: the p-adic norm seems to be differentiable but not continuous)?

- May 10th 2014, 02:10 PMPlatoRe: Non-Archimedian calculus
Are you thinking in these terms ?.

- May 10th 2014, 02:51 PMSlipEternalRe: Non-Archimedian calculus
Yes, the p-adic metric is an example of an ultrametric.

- May 12th 2014, 09:54 AMSlipEternalRe: Non-Archimedian calculus
Never mind. I see what I was doing wrong. The p-adic norm is neither continuous nor differentiable at 0. So, differentiability still implies continuity.

- May 12th 2014, 10:38 AMSlipEternalRe: Non-Archimedian calculus
I spoke too soon. The p-adic norm was just a poor example. Instead, consider defined by (where is the set of p-adic numbers, and is the p-adic norm of ).

First, I will consider continuity at 0. Consider the sequence . This sequence converges to 0 with respect to the p-adic metric, but for all while . Hence, is not continuous at . Yet:

Hence, the derivative exists at . So, differentiability does not imply continuity. Did I make a mistake somewhere?

Edit: In general, if , then if , , so

Hence, agrees with the derivative found for , as well. - May 12th 2014, 11:57 AMSlipEternalRe: Non-Archimedian calculus
Ohhh!!! I see what I am doing wrong. I am looking at convergence in the reals rather than the p-adics. as , so does not exist. My mistake.