Marc Olschok wrote: >quasi 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.
In the Calculus context, the missing point scenario is classified as either a removable discontinuity or a non-removable discontinuity depending on whether the relevant limit exists.
In particular, the function
f(x) = (x^2 - 9)/(x - 3)
is said to have a removable discontinuity at x = 3 since
(1) f is not defined at x = 3.
(2) lim (x -> 3) f(x) exists.
The level of discussion is key here.
At the Calculus level, precisely because they want to discuss the concept of removable versus non-removable discontinuities, the question
"Is f continuous at x = 3?"
is not regarded as a meaningless question.
As a reference, if you have access to the text
Stewart Calculus - Early Transcendentals, 6th Ed (2008)
On page 119, continuity at x = a is defined this way:
f is continuous at x = a if lim (x -> a) f(x) = f(a)
On the same page, the definition is recast as 3 requirements:
f is continuous at x = a if
(1) f(a) exists (2) lim (x -> a) f(x) exists (3) lim (x -> a) f(x) = f(a)