Arturo Magidin wrote: > >A real valued function of real variable f(t) is >"continuous at x=a" if and only if: > >(i) It is defined at a; >(ii) The limit of f(t) as t approaches a exists; and >(iii) the value of the function at a equals the value of >the limit of f(t) as t approaches a. > >A real valued function of real variable f(t) is "continuous >at all real numbers" or "continuous everywhere" if and only if >it is continuous at x=a for all real numbers a.
In the Precalculus/Calculus context, the above is consistent with the definitions I'm familiar with.
On the other hand, I've never seen the usage you describe below, at least not in the current context.
>A real valued function of real variable f(t) is "continuous" >if and only if it is continuous at all points in its domain.
In the Precalculus/Calculus context, here is my take ...
A function f is continuous on an interval if it is continuous at all points of that interval (with one-sided continuity acceptable if the interval has an endpoint).
A function f is "continuous" if the domain of f is an interval and f is continuous on that interval.
The function x^2 is continuous since its domain is the interval (-oo,oo) and it's continuous on (-oo,oo).
The function sqrt(x) is continuous since its domain is the interval [0,oo) and it's continuous on [0,oo).
The function ln(x) is continuous since its domain is the interval (0,oo) and it's continuous on (0,oo).
The function 1/x is _not_ continuous since its domain is not an interval. Of course one _can_ say that 1/x is continuous on the each of the intervals (-oo,0) and (0,oo).
The function tan(x) is _not_ continuous since its domain is not an interval. Of course one can say that tan(x) is continuous on the interval (-Pi/2,Pi/2) and more generally tan(x) is continuous on any interval of the form (a - Pi/2, a + Pi/2) where a = k*Pi for some integer k.
By your usage, all of the above functions, including the functions 1/x and tan(x) would be regarded as "continuous", without qualification.
I've never seen that usage.
Worse, it jars with the naive, Precalculus concept of continuity, typically expressed as:
"A function is continuous if the graph has no breaks."
"One can trace the graph of the function without lifting the pen from the paper."
>So the function f(t) = (t^2-9)/(t-3) is continuous;
Can you find a reference to a textbook at the Precalculus or Calculus level which has a definition by which a function similar to the function f above would be regarded as a continuous function?
>it is not, however, continuous at all real numbers.
No problem with that.
After all, f(3) is not defined, so f is certainly not continuous at t = 3.
The issue is whether, in the Precalculus/Calculus context, there is any standard convention by which the function