Re: Clarification on definition of limits

Randy Poe wrote:

Stephen J. Herschkorn wrote:


Michael Stemper wrote:


My question is about what happens when the domain of a function is,
for the want of a better term, "coarse". Suppose I define a function,
f(x) = 1.1x, with its domain restricted to the integral multiples of
0.1, and try to find the limit of f(x) as x approaches 0.05.

Intuitively, I'd expect to be able to say that the limit was
(1.1)*(0.05) = 0.055. However, a small neighborhood around 0.05
contains no elements of the domain! That appears to tell me that,
if I want to find the limit at a point not in the domain, all that
I need to do is pick a delta small enough that no points in the
domain are included, and I can claim the limit is anything that
I want it to be, since |f(x) - L| will be smaller than any positive
epsilon for all points in the neighborhood (since there aren't
any).





No, there is nothing wrong - you got it *exactly* right. (Very good!)
We say that the domain in this case is discrete. Any function from a
discrete space is continuous.



Does that mean that differentiation can be defined on
a discrete space?


IIRC, a discrete space is not normalizable, so I don't think one can come up with a useful definition of derivative. In any case, limits are not unique in a discrete space.

What is the point of your question?  You surely already knew the answer.

--
Stephen J. Herschkorn                        sjherschko@xxxxxxxxxxxx
Math Tutor in Central New Jersey and Manhattan

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