continuous functions

[intothevoidx]

Could anyone explain to me how you do this. It says to complete the segmented function so that it is continuous.

 

y={ (x^2-11)/(x+(sqrt11))

---- if x = -----

 

 

 

Thanks

[timo]

You supposedly meant [math] y = \frac{x^2-11}{x+\sqrt{11}} [/math] for [math] x \in R, x \neq -\sqrt{11}[/math] since the function is obviously undefined at x = -sqrt(11). If the value of the function approaches the same value when approaching -sqrt(11) from the right and from the left, you could just define a new function which is equal to the original one for all [math]x \neq -\sqrt{11}[/math] and equals that limit at [math]x = -\sqrt{11}[/math]. I suppose that's what's meant.

[Fred56]

Continuity and limits are probably the first (and toughest) newbie steps into calc, that many make, and it isn't taught that well (as a rule)...

[intothevoidx]

Well, I'm only in precalculus and we haven't gone over limits. Generically speaking, what do I want to do if I encounter any problem that asks me to make the segmented function continuous?

[Fred56]

By "segmented", you are presumably referring to a function which has discontinuities, or steps, or "missing bits", i.e. a discontinuous function? But you understand how a function has an input and an output, a bit like an algorithm.

These, you may also know, are the range and domain (of a function). One's the input (the set of allowed values for x, say) the other's the output (the y values, or eigenvalues -real values). And you should know which is which (don't assume I have told you the "order").

Anyhoo, you need to understand things like Euclidian 2-space, and a zero-dimensional ball (the next down from a circle) and a series that converges (on a limit), and stuff like absolute value, and sets (inclusion and exclusion -open and closed), and commut(ative), and transit(ive). Then it's all pretty plain sailing.

[timo]

Well, I'm only in precalculus and we haven't gone over limits. Generically speaking, what do I want to do if I encounter any problem that asks me to make the segmented function continuous?

I'd propose filling the gaps with sensible values.

[Country Boy]

I wondering exactly what DEFINITION of "continuous" you are using?

[ydoaPs]

I wondering exactly what DEFINITION of "continuous" you are using?

 

It's been a while since I took a calculus course, but doesn't it have to do with [math]\lim_{x\rightarrow{n}}{f(x)}=f(n)[/math] for all values of n which requires the limit of the function to exist over the entire range in which n exists?

[Fred56]

The domain of a function can be open or closed, so therefore it depends on how x or any variable is defined. According to some notes I still have: the domain is every point at which (the definition of) f makes sense. I guess it´s discontinuous otherwise, or maybe assumed to be continuous over some interval (unless otherwise stated).

In input-output terms, if the input is discontinuous, the output may be as well (or if it is, do something about it).

[Country Boy]

It's been a while since I took a calculus course, but doesn't it have to do with [math]\lim_{x\rightarrow{n}}{f(x)}=f(n)[/math] for all values of n which requires the limit of the function to exist over the entire range in which n exists?

That's certainly plausible but I was specifically asking Intothevoidx what definition of continuous HE was using since he is asking about continous functions but tells us he hasn't "gone over limits". What is his DEFINITION of "continuous" if it does not involve limits?

[ydoaPs]

That's certainly plausible but I was specifically asking Intothevoidx what definition of continuous HE was using since he is asking about continous functions but tells us he hasn't "gone over limits". What is his DEFINITION of "continuous" if it does not involve limits?

 

uhhh, no breaks in the graph

[Fred56]

Here's a discontinous function: [math] f(x) = \frac 1 {|x-1|} [/math]

 

The domain (input) of [math]f [/math] is: [math] x-1[/math] if [math]x-1 \ge 0

[/math] and: [math] -(x-1) [/math] if [math] x-1 < 0 [/math]; and excludes (is closed to) the set {1} (from the real numbers).

So [math]f [/math] is discontinous, about the value 1, at which it's an asymptote (x can be any value less than or greater than 1, but not equal).

You can define the domain of any function to itself be discontinous, or "patch" things so the range (output) isn't...

[Country Boy]

uhhh, no breaks in the graph

 

Then GRAPH the function! It should look awfully familiar. Probably, if you use a graphing calculator, it will completely ignore the "difficulty". Just check what value the graph gives at the "difficult" point.

 

The domain of a function can be open or closed, so therefore it depends on how x or any variable is defined. According to some notes I still have: the domain is every point at which (the definition of) f makes sense. I guess it´s discontinuous otherwise, or maybe assumed to be continuous over some interval (unless otherwise stated).

In input-output terms, if the input is discontinuous, the output may be as well (or if it is, do something about it).

You can define the domain of any function to itself be discontinous, or "patch" things so the range (output) isn't...

 

I'm sorry but this makes no sense to me. You seem to be defining "continuous" or "discontinuous" to be a property of the DOMAIN of a function rather than the function itself- which it definitely is not.

 

The function f(x)= 0 is x<= 0, 1 if x> 1 has domain all real numbers but is discontinuous at x= 0.

 

The function f(x)= 0 if x is rational, 1 if x is irrational, has domain all real numbers but is discontinuous for all x.

 

The function f(x)= 1/n if x is rational and x= m/n reduced to lowest terms, 0 if x is irrational, has domain all real numbers but is discontinuous for all x EXCEPT x= 0.

[Fred56]

You seem to be defining "continuous" or "discontinuous" to be a property of the DOMAIN of a function

Perhaps I should have said (I thought I did) that the domain can be discontinous -e.g.: -12 <= x < 0, 0 < x <= 224 which is a (closed) domain for x which has a discontinuity at 0. The range of a function depends on the domain (and of course, the function), but there may be a function which is continuous over its entire range with the domain (for x) specified in the first sentence of this post...

Discontinuity is a general concept you can apply to any domain or range, it isn't restricted to the output of a function. Look at the function again in the post previous to your last, the domain (x) is discontinuous (excludes 1), and the range is discontinuous also.

 

P.S. Think about how a function's domain can be the range of another function, etc.

[Country Boy]

Perhaps I should have said (I thought I did) that the domain can be discontinous -e.g.: -12 <= x < 0, 0 < x <= 224 which is a (closed) domain for x which has a discontinuity at 0.

Then "continous" is the wrong word. Functions are continous or discontinuous. Sets (domains) are "connected" or "non-connected".

 

 

The range of a function depends on the domain (and of course, the function), but there may be a function which is continuous over its entire range with the domain (for x) specified in the first sentence of this post...

Discontinuity is a general concept you can apply to any domain or range, it isn't restricted to the output of a function. Look at the function again in the post previous to your last, the domain (x) is discontinuous (excludes 1), and the range is discontinuous also.

You may me translating from another language in which the same terms are used but, again, in English, "continuous" only applies to functions, "connected" to sets.

 

P.S. Think about how a function's domain can be the range of another function, etc.

I don't see what that has to do with the problem.

[Fred56]

Sets (domains) are "connected" or "non-connected"

So my calculus lecturer (who has a graduate degree), is wrong to say that a function's domain can be discontinuous, then? I must see what he has to say about this.

P.S. Think about how a function's domain can be the range of another function' date=' etc.

 

I don't see what that has to do with the problem.[/quote']

If not, then you should consider the possibility that you still don't understand what a mathematical (or any other kind of) function is, possibly. I recommend a chat with a math lecturer (one with a degree, maybe).

[iNow]

So my calculus lecturer (who has a graduate degree), is wrong to say that a function's domain can be discontinuous, then? I must see what he has to say about this.

 

If not, then you should consider the possibility that you still don't understand what a mathematical (or any other kind of) function is, possibly. I recommend a chat with a math lecturer (one with a degree, maybe).

 

Fred - That's both an appeal to authority and an ad hominem. Why not validate with the instructor that you heard them correctly, or show why and where HallsofIvy is wrong.

 

Of course, you don't have to, but I know that at least I would take you more seriously if you did. Cheers mate.

[Fred56]

Well, if it's true that a function can't be described as having a continuous domain, you have to describe a "connected set" instead is just saying "set terminology is how you 'have' to say it". Which is just being precious with language.

Otherwise how can any function have the output (i.e. a function) of something as an input; otherwise is the word "continuous" suddenly only useful in a certain sense, or somesuch (completely meaningless) thing?

Consideration: it is more precise to say that the function (i.e. the range) is continuous over a domain, but again, this doesn't mean you can't say that the domain is too, and, there's the obvious objection to "you can't call the domain of a function continuous", that a domain can be, as stated, the range of a function. Especially the function of numbers, Euler's one which is it's own derivative, for a start.

[Country Boy]

Well, if it's true that a function can't be described as having a continuous domain, you have to describe a "connected set" instead is just saying "set terminology is how you 'have' to say it". Which is just being precious with language.

No, that's using language correctly. Something you have to be very careful about in mathematics.

 

Otherwise how can any function have the output (i.e. a function) of something as an input; otherwise is the word "continuous" suddenly only useful in a certain sense, or somesuch (completely meaningless) thing?

I really can make no sense out of that sentence.

 

Consideration: it is more precise to say that the function (i.e. the range)

"function (i.e. the range)"? No, a "function" is distinct from its range.

 

is continuous over a domain, but again, this doesn't mean you can't say that the domain is too, and, there's the obvious objection to "you can't call the domain of a function continuous", that a domain can be, as stated, the range of a function. Especially the function of numbers, Euler's one which is it's own derivative, for a start.

Yes, of course. ANY set can be the domain or the range of a function. In any case, in mathematical analyis, the word "continuous" is defined only for functions, not for sets.

[Fred56]

No, a "function" is distinct from its range.

How is it 'distinct' though, since we're being careful here?

Let's say that some interval of the real number line is the domain of a function called f: how is this domain of f not itself a function or the range of some function (what the hey, call it f')?

[Country Boy]

How is it 'distinct' though, since we're being careful here?

Let's say that some interval of the real number line is the domain of a function called f: how is this domain of f not itself a function or the range of some function (what the hey, call it f')?

I never said the domain of some function, f, cannot be the range of another function- since both domain and range are sets. that's quite possible. My point was that a function is NOT its domain or range.

 

Earlier, when I said that I did not see what your comment about the domain of one function being the range of another had to do with the question of whether the word "continuous" applied to the function or domain, you said, "If not, then you should consider the possibility that you still don't understand what a mathematical (or any other kind of) function is, possibly. I recommend a chat with a math lecturer (one with a degree, maybe)."

Actually, I talk to people with math degrees every day- I have a Ph.D. myself and have been teaching college math for about 30 years. I know very well what a function is as well as the concepts of "domain" and "range" of a function:

 

A "function" is a set of ordered pairs with the property that no two pairs have the same first member. The "domain" of a function is the set of all first members in those pairs, the "range" is the set of all second members (x and y if you like). It is common to specify the domain and give some rule by which the "y" corresponding to a given "x" in the domain. It is also common to just give the "rule" with the understanding that the domain is the set of values to which the rule can be applied. However, the "domain" and "range" of a function are distinct from the function itself- that has been my point all along.

[iNow]

A "function" is a set of ordered pairs with the property that no two pairs have the same first member. The "domain" of a function is the set of all first members in those pairs, the "range" is the set of all second members (x and y if you like). It is common to specify the domain and give some rule by which the "y" corresponding to a given "x" in the domain. It is also common to just give the "rule" with the understanding that the domain is the set of values to which the rule can be applied. However, the "domain" and "range" of a function are distinct from the function itself

Thank you. That was incredibly clear and concise.

[Country Boy]

Well, it's practically a quote from any basic textbook. I have been known to glance at one occassionally!

[Fred56]

Thank you. That was incredibly clear and concise

Except that it's also perfectly ok to say that a function is it's output, and depends on its input. It's also fairly acceptable (where I come from), to say that because the input of a function can be the output of a function (what the hell, let's call it the range of another function, for accuracy), then they can be the same thing, in that sense. I dare you to define the domain of any function as not itself being the range of some other function, also. Even using set notation. And let's not forget Eulers series which has range and domain identical (what does identical mean again?)

However, the "domain" and "range" of a function are distinct from the function itself

P.S. You still haven't described how a function is 'distinct' from its range , beyond that it's meant to be the black box -the rule that outputs something (but you have described the set of duples that represent the domain and range)

Or maybe (one might) just ignore the black box and look at the output, say, and call it the same thing. This is ok to do, unless you actually want to look inside the box. So the function is what makes it 'tick', so to speak, but it's also (if you study electronics or EE), ok to equate with its output (the ticking). In other words "what it does" can mean both what it outputs (and therefore inputs), and what the internal workings are...

This being a math thread, I obviously should have left my engineering glasses behind.

Then again, looking at mathematical functions and electronic circuits (especially 'passive' ones), I think lends the electronic circuit model some clout in a math classroom, maybe. Even a small bulb attached to a battery is a 'function', or illustrates what a function is...

 

P.P.S.

I have a Ph.D. myself and have been teaching college math for about 30 years

Congratulations. I don't, but I have met students in their 3rd year of a CS degree who don't know the difference between a verb and a noun in their native language (one of them asked me to explain it)...

[Country Boy]

Except that it's also perfectly ok to say that a function is it's output, and depends on its input. It's also fairly acceptable (where I come from), to say that because the input of a function can be the output of a function (what the hell, let's call it the range of another function, for accuracy), then they can be the same thing, in that sense. I dare you to define the domain of any function as not itself being the range of some other function, also. Even using set notation. And let's not forget Eulers series which has range and domain identical (what does identical mean again?)

You keep asserting that "the domain of one function can be the range of of another". I've never said they can't! I have agreed with that several times now. I also have never suggested that a function cannot have range and domain identical- f(x)= x is a good example. I have been refusing to accept your assertion that "a function is its output". They are not at all the same thing! I notice you appended ", and depends on its input". That's precisely my whole point- the function is exactly HOW the "output" depends on the "input"!

 

P.S. You still haven't described how a function is 'distinct' from its range , beyond that it's meant to be the black box -the rule that outputs something (but you have described the set of duples that represent the domain and range)

Then I don't know what more to say. That set of duples does a lot more than "represent the domain and range".

The functions f: {(0,1), (1,1), (2, 2), (3, 3)} and g: {(0, 3), (1, 2), (2, 1), (3, 1)} have exactly the same Domain: {0, 1, 2, 3} and exactly the same range, {0, 1, 2, 3}, which are themselves identical, but are very different functions!