Continuity of square root of a function

[MrPanda]

Been getting different answers to the following question:

Given f(x) = sqrt ( x^2 - 25 ), it is continuous on: ... a) none of these ... b) [ 5, + inf ) ... c) ( - inf, -5 ] U [ 5, + inf ) ... d) ( -inf, -5 ) ... e) ( -5, 5 )

inf = infinity cont = continuity / continuous

I believe the correct solution is c), but have been told it is ...... a) None of These , by a couple of people. What do you think ??

My reasoning: Calculus texts state that root functions, like square root, are continuous at every point in their domain, [ one even stated this as a Theorem ] and the domain for this problem is x >= 5 or x < = -5 which is the same as c) above , as U means "or" ........ [ BTW .. Wolfram Alpha program also states this problem is cont over it's domain ! ].

Others argue that the problem is at the endpoints 5, -5 , and what happens on the other side of those values, so I guess they feel c) would be correct if the solid [ , or ) instead.......

From several Calculus texts, the continuity at an endpoint need only be right hand limit , or left hand limit, whichever applies there, and the limits both exist and = 0 for approaching 5 , or - 5 from their right, or left as needed. [ The text refers to this as continuous from the right, and from the left ].

An example given in a Calculus text is sqrt( 1 - x^2) , where they show continuity for ( -1, 1 ) then consider Rt./ Left hand limits to confirm cont. at x = + or - 1, to arrive at cont on [ -1, 1 ] as the answer .... .....they don't seem to care what takes place on the "other side" of -1 , +1 , as some have argued on my stated problem above. I don't believe the intervals being split in my answer c) , can be an issue, as several other continuity problems in the same text has cont intervals such as ( - inf, 2 ) U ( 2, inf ) for answers on a point discontinuity at x = 2.


Again...your thoughts / analysis ??

 

[Romsek]

It's none of them.

[MATH]f(x) = \sqrt{x^2 - 25}[/MATH] has a discontinuity at [MATH]x=\pm 5[/MATH] and thus these points cannot be included in any domain on which \(\displaystyle f(x)\) is continuous.

[MATH]f(x)[/MATH] is continuous on [MATH](-\infty, -5) \cup (5,\infty)[/MATH]

 

[Dr.Peterson]

Been getting different answers to the following question:

Given f(x) = sqrt ( x^2 - 25 ), it is continuous on: ... a) none of these ... b) [ 5, + inf ) ... c) ( - inf, -5 ] U [ 5, + inf ) ... d) ( -inf, -5 ) ... e) ( -5, 5 )

An example given in a Calculus text is sqrt( 1 - x^2) , where they show continuity for ( -1, 1 ) then consider Rt./ Left hand limits to confirm cont. at x = + or - 1, to arrive at cont on [ -1, 1 ] as the answer .... .....they don't seem to care what takes place on the "other side" of -1 , +1 , as some have argued on my stated problem above. I don't believe the intervals being split in my answer c) , can be an issue, as several other continuity problems in the same text has cont intervals such as ( - inf, 2 ) U ( 2, inf ) for answers on a point discontinuity at x = 2.

Again...your thoughts / analysis ??

Since you have found that sources differ, the real question is, what does YOUR textbook (and/or instructor) say about continuity at endpoints? Whoever is judging your answer gets to decide.

 

[MrPanda]

It's none of them.

[MATH]f(x) = \sqrt{x^2 - 25}[/MATH] has a discontinuity at [MATH]x=\pm 5[/MATH] and thus these points cannot be included in any domain on which \(\displaystyle f(x)\) is continuous.

[MATH]f(x)[/MATH] is continuous on [MATH](-\infty, -5) \cup (5,\infty)[/MATH]
I actually got a Mathematician from a local university [ PHD ] to help answer and they said that None of these was incorrect, and my solution was the correct one. The Stewart Calculus text, as well as Larson Text also has several Thm's and a problem or two that seems to indictae that a problem like this, +5, -5 are actually ok as they are one sided limits that fit the definition of continuity at an end point, and that root functions are continuous over their domains, and composite functions, as in this case, are continuous at these values.....so i'm going to have to go along with them. Thanks Anyway.

 

[Dr.Peterson]

As I said, mathematicians differ on their definitions in this area, so the "correct" answer depends on your own context. You have not indicated who is going to judge your answer! Is this for a class you are taking, or for an admissions exam, or what?

 

[JeffM]

To expand a bit on what Dr. Peterson has said, this depends on definitions. If we define continuity as equivalent to

[MATH]f(x) \text { is continuous at } a \iff[/MATH]
[MATH]f(a) \in \mathbb R \text { and } \lim_{x \rightarrow a^+} f(x) = f(a) = \lim_{x \rightarrow a^-}f(x)[/MATH],

then your function is not continuous at - 5 (because it has no limit from the right there) and is not continuous at 5 (because it has no limit from the left there).

Some texts give this definition in two stages. They first give a definition of limit specifying that the limit exists if and only if both the right and left limits are real and equal. Then they define f(x) as continuous at a if and only if f(a) is real and equals the limit of f(x) as x approaches a.

With that kind of definition, it is easy to confuse statements about existence and about continuity. Your function exists at 5 and - 5 so the the domain of f(x) is everything except (- 5, 5), but the function is continuous only if x < - 5 or x > 5.

 

[Dr.Peterson]

To be more specific, the Larson book I have at hand has a "Definition of Continuity on a Closed Interval", that says, "A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (x,b) and ... is continuous from the right at a and continuous from the left at b." It doesn't actually say that the function is continuous at a and b; this is only a special definition of continuity on an interval, and is presumably given only to make it easier to talk about such situations.

 

[MrPanda]

Update: Here's what I've discovered.

The solution I posted is now confirmed to be the correct one.

1. Two University Math Prof. [ 1 a Co -Author of our text ], have responded to my e-mail problem and confirmed my solution.

2. 2 Calculus texts I located in the Library [ out of Dozens and Dozens !! ] finally had problems like this. One worked out a problem with the sqrt( x^2 - 1 ) in detail, one as a HW problem and solution with a similar problem, also confirm that my solution and the continuity issue at the endpoints, as the solutions would be the same had they used my x^2 - 25 instead.

3. Wolfram Alpha calculators [ continuity and discontinuity ] state the problem is continuous over it's Domain, which includes the endpoints, as well as Desmos.
In all cases, there seems to be no issues at the endpoints for continuity on open, closed, or 1/2 open intervals.

I just needed to confirm the given solution was in fact incorrect, and I was correct, before I let the fellow Math teacher know about the incorrect solution on her Final Exam.....and I kept getting different answers from people , making me wonder if I missed something critical.... that appears not to have happened.

Mr Panda
{ degrees in Math & Physics }

 

[Dr.Peterson]

You've never actually told us the context of your question, which can be essential.

I think you are saying that this is about a test for a course you are teaching, in which a colleague disagreed with your textbook.

For that purpose, all you had to do was refer to what that textbook said; you don't need to find unanimity around the world (which does not always exist -- this is one of the downsides of students having access to the internet). If the textbook doesn't even mention this kind of question, then the problem should not be on a test in the first place.

But at least two books I've checked agree that a function is considered to be continuous over an interval when it is continuous from one side at its boundaries. The most likely issue others would have is if they are approaching the problem from a different context (a higher level course), or are not given the exact wording of the question, which must distinguish continuity on an interval from continuity at every point in that interval.

 

[Steven G]

Many years ago while I was adjuncting at a number of different college I stayed far far away from this type of problem as the textbooks i was using did not agree on the definition of continuity over a closed interval. I would not say for a moment that some of the books had the wrong definition.
The answer to the problem you stated is clearly dependent on the textbook/instructor.