On 10/20/2013 3:54 AM, quasi wrote: > Hetware wrote: >> quasi wrote: >>> >>> On the other hand, suppose your proposed question is reworded >>> in the following way ... >>> >>> Given: >>> >>> f(x) is defined and continuous for all x in R. >> >> I believe we can drop the "defined" since it is implied >> by "continuous". > > Yes. > >>> Then if g is defined by >>> >>> g(x) = limit (s -> x) f(s) >>> >>> can g(x) be used to determine the value of f(x) for all x in R? >>> >>> The answer is yes -- in fact, f(x) = g(x) for all x in R. >> >> What is your opinion of the definition given as follows: >> >> Let f(x) be continuous over R, and f(x) = x/x for all x in R >> where x/x is a determinate form? > > I wouldn't use the phrase: > > "where x/x is a determinate form" > > Instead say: > > "where x/x is defined" > > You could also say it more simply this way: > > "Let f be a function which is continuous over R and such > that f(x) = x/x for x != 0." > > Or even simpler: > > "Let f(x) = 1. > > But at this point, you surely understand the terminology and > conventions relating to this issue as used in the text by > Thomas. Why not just go on from there? > > quasi >
I finally resolved my confusion. I was treating the proposition that f is continuous to be equivalent to defining f to be continuous.
The reason I spent so much time with this is because my mind automatically removed the removable discontinuity when I saw the definition f(t) = (t^2-9)/(t-3). I needed to understand why I could see f(t) as continuous. I now understand that I was implicitly imposing additional conditions on f(t).
I had come to that conclusion about a week ago, then I tried to formalize my understanding, and got confused with the distinction between a proposition and a definition.
It didn't help that others were insisting that I could not assert continuity as part of a definition.
This was posted elsewhere in this thread earlier today. It's how I finally resolved my confusion.
Let P(f) <=> f(x) is continuous for all x in R. Let D(f) <=> f(x) has a finite value for all x in R. Let L(f) <=> limit[f(s), s->x] exists for all x in R. Let Q(f) <=> f(x) = limit[f(s), s->x]
P <=> D & L & Q Q => D
Let f(x) be in R for all x in R other than x=0, f(0) = 0/0 and limit[f(s), s->x] exist for all x in R.
D(f) => F L(f) => T Q(f) => F !P(f) <= F & T & F
Let g(x) be a continuous function on R such that g(x) = f(x) when f(x) is a real number.
D(g) => T L(g) => T Q(g) => T P(g) <= T & T & T
So quasi was correct. I can define a function g(x) to be continuous, but it is not the same function as the function f(x) used to define g(x) everywhere f(x) is defined.