Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
Re: Is (t^2-9)/(t-3) defined at t=3?
Posted: Oct 8, 2013 4:36 PM

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.

Examples:

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."

or equivalently,

"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;

Very strange.

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

f(t) = (t^2 - 9)/(t - 3)

would be called a continuous function.

I don't think so.

quasi

Date Subject Author
9/28/13 Hetware
9/28/13 Michael F. Stemper
9/28/13 scattered
9/28/13 Hetware
9/28/13 quasi
9/28/13 Hetware
9/28/13 quasi
9/28/13 Peter Percival
9/29/13 quasi
9/28/13 Hetware
9/28/13 Richard Tobin
9/28/13 Hetware
9/28/13 tommyrjensen@gmail.com
9/29/13 Hetware
10/6/13 Hetware
10/6/13 Peter Percival
10/6/13 Hetware
10/6/13 quasi
10/8/13 quasi
10/7/13 Peter Percival
9/29/13 Michael F. Stemper
9/29/13 Hetware
9/29/13 quasi
9/29/13 Hetware
9/29/13 magidin@math.berkeley.edu
10/6/13 Hetware
10/6/13 magidin@math.berkeley.edu
10/7/13 Hetware
10/7/13 LudovicoVan
10/7/13 Peter Percival
10/8/13 magidin@math.berkeley.edu
10/12/13 Hetware
10/12/13 fom
10/13/13 magidin@math.berkeley.edu
10/13/13 Richard Tobin
10/13/13 Hetware
10/13/13 Peter Percival
10/13/13 fom
10/13/13 magidin@math.berkeley.edu
10/13/13 magidin@math.berkeley.edu
10/8/13 quasi
10/8/13 magidin@math.berkeley.edu
10/8/13 quasi
10/8/13 quasi
10/12/13 Hetware
10/13/13 quasi
10/13/13 Peter Percival
10/9/13 magidin@math.berkeley.edu
10/9/13 fom
10/10/13 magidin@math.berkeley.edu
10/10/13 fom
10/7/13 Peter Percival
10/7/13 Hetware
10/7/13 fom
10/7/13 Peter Percival
9/29/13 quasi
9/30/13 Peter Percival
9/30/13 Peter Percival
9/30/13 Peter Percival
9/30/13 RGVickson@shaw.ca
9/30/13 Roland Franzius
9/30/13 Richard Tobin
9/30/13 RGVickson@shaw.ca
9/28/13 Peter Percival
9/28/13 Hetware
9/29/13 Peter Percival
9/28/13 Virgil
9/29/13 quasi
9/29/13 Virgil
9/29/13 Hetware
9/29/13 quasi
9/29/13 Hetware
9/29/13 LudovicoVan
9/29/13 quasi
9/29/13 Virgil
9/29/13 magidin@math.berkeley.edu
9/29/13 Peter Percival
9/29/13 FredJeffries@gmail.com
9/30/13 Hetware
9/30/13 magidin@math.berkeley.edu
10/6/13 Hetware
10/6/13 Peter Percival
10/6/13 Peter Percival
10/6/13 magidin@math.berkeley.edu
10/6/13 Peter Percival
10/6/13 magidin@math.berkeley.edu
10/6/13 David Bernier
9/29/13 Peter Percival
9/28/13 Hetware
9/29/13 Richard Tobin
9/30/13 Ciekaw
9/30/13 Robin Chapman
9/30/13 Virgil
9/30/13 LudovicoVan
9/30/13 LudovicoVan
10/6/13 Hetware
10/7/13 Robin Chapman
10/7/13 David Bernier
10/7/13 Hetware
10/7/13 LudovicoVan
10/8/13 Hetware
10/9/13 Peter Percival
10/9/13 Richard Tobin
10/7/13 Peter Percival
10/8/13 Hetware
10/8/13 Virgil
10/8/13 Hetware
10/9/13 magidin@math.berkeley.edu
10/9/13 Peter Percival
10/10/13 Ciekaw
10/9/13 Peter Percival
10/10/13 Tim Golden BandTech.com
10/13/13 Hetware
10/13/13 Peter Percival
10/13/13 Hetware
10/14/13 Peter Percival
10/13/13 Hetware
10/13/13 fom
10/13/13 Hetware
10/13/13 fom
10/14/13 fom
10/14/13 Hetware
10/14/13 magidin@math.berkeley.edu
10/14/13 magidin@math.berkeley.edu
10/14/13 Peter Percival
10/14/13 Hetware
10/14/13 quasi
10/16/13 @less@ndro
10/16/13 quasi
10/19/13 Hetware
10/19/13 quasi
10/19/13 Hetware
10/20/13 fom
10/20/13 quasi
10/20/13 Hetware
10/20/13 fom
10/20/13 Hetware
10/20/13 Peter Percival
10/20/13 Richard Tobin
10/20/13 Hetware
10/30/13 @less@ndro
10/19/13 Hetware
10/10/13 Ronald Benedik
10/10/13 Peter Percival
10/10/13 Virgil
10/18/13 Hetware
10/19/13 Peter Percival
10/19/13 fom
10/19/13 Peter Percival
10/19/13 Hetware
10/19/13 Peter Percival
10/19/13 Hetware
10/19/13 fom
10/19/13 magidin@math.berkeley.edu
10/19/13 Hetware
10/19/13 magidin@math.berkeley.edu
10/20/13 Hetware
10/20/13 quasi
10/20/13 quasi
10/20/13 Hetware
10/20/13 Peter Percival
10/20/13 magidin@math.berkeley.edu
10/20/13 Hetware
10/20/13 Arturo Magidin
10/20/13 Hetware
10/20/13 magidin@math.berkeley.edu
10/19/13 fom