Re: Clarification on definition of limits

Michael Stemper wrote:

I recently learned that the definition of the limit of a function has
the clause "for all x in the domain of f" in it, something that I'd
always glossed over before. This means that a function could, for
example, be defined only on the rationals, but still have (AIUI) a
limit defined for x->pi.

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).

Presumably, there's some subtlety that I'm overlooking here that
prevents such an unsatisfactory state of affairs. It can't be as
simple as "the limit's only defined for points in the domain of
the function". What is it?


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.

You may want to take a look at general metric spaces or, even better, general topology.

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

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