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Topic: Is (t^2-9)/(t-3) defined at t=3?
Replies: 166   Last Post: Oct 30, 2013 9:41 AM

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@less@ndro

Posts: 212
Registered: 12/13/04
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 16, 2013 11:37 AM
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quasi <quasi@null.set> wrote:
> As I've previously suggested, the wording is key ...
>
> Consider the following problems ...
>
> Problem (1)
>
> Let f(x) = (x^2 - 9)/(x - 3).
>
> (a) Is f(3) defined?
>
> answer: No.
>
> (b) Is f continuous on its domain?
>
> answer: Yes.
>
> (c) Is f continuous on the set of real numbers?
>
> answer: No, f is not continuous at x = 3.

I have not read everything in this (IMHO broken) thread, so this might
already have been pointed out. The problem is that question (c) and its
answer is meaningless, once it is settled that 3 is not in the domain
of f. One might as well ask if f is continuous on the quaternions.
The above map f: R \ {3} --> R is continuous (in the sense of (b)).
No need for further complications.

The main point is that the domain of a map (as well as its codomain)
is part of its specification. So for example

f: R \ {3} --> R with f(x) = x+3
and
g: R --> R with g(x) = x+3

are two different maps, despite having the same function term.
This distinction should be made clear from the start in any
modern textbook worth the name. It is understandable that, when dealing
with e.g. rational functions on the real line, analysts just give
the function term without further comment, because they already know
what they are doing, unlike beginning students. I am also aware that
poles of rational functions are sometimes called "points of discontinuity",
but if students are taught precise definitions for maps from the
beginning (i.e. before taking calculus), this (slighly misleading)
wording should not derail them.

>
> (d) Does lim (x->3) f(x) exist?
>
> answer: Yes, the limit is 6.
>
> (d) Does there exist a function g continuous on the set of
> real numbers such that g(x) = f(x) for all x != 3?
>
> answer: Yes, the function g(x) = x + 3 has that property,
> and in fact, it's the only such function.
>[...]


--
Marc


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9/28/13
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10/16/13
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10/19/13
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