On 10/16/2013 5:38 PM, quasi wrote: > 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. > > Not really. > > 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. > > but > > (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) > > On page 120, the following example is given ... > > ======================================================= > > Question: > > Let f(x) = (x^2 - x - 2)/(x - 2). > > Where is f discontinuous? > > Answer: > > f is discontinuous at x = 2 since f(2) not defined. > > ======================================================= > > quasi >
Of course, when taking a calculus test which assumed the widely accepted definitions and interpretations, I gave the answer you have provided.
Let me propose a test question:
Given f(x) defined over R, limit[f(s), s->x] exist over R, and g(x) = limit[f(s), s->x] can g(x) be used to find the value of f(x) for all x in R?