For you canNOT extend the function tan x to have domain R and still be continuous. This is obviously wrong. Moreover, this 'definition' is limited to R and other linearly ordered topological spaces , while continuity is a much more generally applicable property.
For more details see Ron Bruck's post. Brian M. There is a lot of disagreement and outright sloppiness in the textbook circles about it. Also, according to this definition, a function would be discontinuous at every isolated point of its domain, which is contrary to the more general topological definition of continuity.
That's why I reject the above definition. One can amend the definition of continuity of f by adding "or, if c is an isolated point of the domain of f", but this looks forced. To put isolated and non-isolated "accumulation" points under one roof, to call f continuous at c, we require that c be in the domain of f as above , and the inverse image of every neighbourhood of f c contain a neighbourhood of c, relative to the domain of f.
Look up the missing definitions in a real analysis book, or wait for the course. Some want to distinguish "removable" and "unremovable" discontinuities, depending on whether the limit exists or not, but that is another topic.
I have seen other confusing and confused attempts at the definition of limits and continuity in textbooks , but I'd rather not clutter my response with them. All this is meant to be discussed in the context of real numbers, in their usual topology, i. But that would be another article. One-sided infinities will not help here.
Cheers, ZVK Slavek. This definition is good for basic calculus courses, while it intesifies the learning of the concept of limit, taught about the same time. This definition is good in a course heading for topology. This latter definition is conceptual and cannot be checked by a pocket calculator, to the satisfaction of a lazy student.
Lounesto need not be told the facts; I am addressing a wider audience. Some time ago, I was lured into a dispute about "definitions for the masses vs. The other debater turned out to be trolling, but I digress. I am still convinced that holding back a part of the definition which is so easy to formulate would be cheating: [ On second thought, it is not so forced if we draw attention to the fact that limits are unavailable or not unique, depending on the definition of a limit at isolated points, hence a default clause in in order.
This latter. The impression about a single sequence being enough , for less informed readers, is false. We know, of course, that the limit is 0. How do I prove it without the hand-waving L'Hopital style manipulation? For continuity, one drops the condition "but different from c". And: isolated points are covered by Heine without any trouble. The neighborhood definition is easy to illustrate graphically, both by static and by moving pictures.
It is also an opportunity to put the student "in the driver's seat": instead of sitting back and watching teeny-weeny cobolds pushing x towards 3 with all their might, but never reaching it, we answer a challenge: tell me the tolerance - how close you want f x to be to L, the proposed limit or to f c for continuity - and I will specify how close x should be to c.
Engineering students would or should relate to that. Regards, ZVK Slavek. What is the current definition of tan x as it's obviously different from the version I learned at school! The domain of tan x is R and so tan x has discontinuities.
In what follows, I'm going to take "discontinuities" to mean "points of discontinuity". The convention for algebraic and trigonometric functions is that when we write a formula f x for such a function, the domain is the set of x for which f x is well-defined. Sometimes we mean over the reals, sometimes over the complexes. That's its domain, not R. You can't plot tan on R, because it isn't defined there. On its domain it is continuous. And that's the definition of continuous function; a function is continuous iff it is continuous at every point of its domain.
Functions don't "know" anything about points outside of their domain. The continuity of a function is solely a property of the function, another property of which is its domain. One may define continuity by limits ugh! This function doesn't know about the rest of R. OK, this is a whimsically extreme remark. The reals play a special role here: students can "see" the discontinuity when they look at the graph, and they think we're weird for calling such a function continuous.
But it's easy to call such points what they are: poles. And to take the opportunity to introduce students to series, and an explanation of what tan x looks like in the neighborhood of a pole, by dividing the series for cos into the series for sin.
The domain of the tangent function can't be R, since tan x isn't defined on all of R. Jeremy's question is legitimate: consider a rope of one meter long and another rope of the same length with the middle point missing. Are they both continuous ropes?
If mathematicians say "yes", it is legitimate to ask: do mathematicians serve the real world? I think the problem you're having is not with the definition of tan but with the definition of domain. Let me take a whack at this; doubtless someone else will put it in more formal language. The domain of a function is the set of allowable inputs to that function, or the set of numbers for which the function is defined. Graphically, the domain corresponds to the set of points on the x axis that are under or over points on the graph of the function -- in other words, the projection of the function onto the x axis.
You can't assume that tan has the same domain as sin and cos, even though it's defined in terms of those functions. Because when cos x is 0, tan x is undefined.
Therefore those values are not in the domain of the tan function. The courtesy of providing a correct reply address is more important to me than time spent deleting spam. It is not reasonable to expect "continuous" to mean the same thing in math as in real life. After all, do we complain because there is no grass growing in the "field" of real numbers?
One may define continuity by limits. The standard epsilon-delta definition is the one I've usually seen. Get as close as you like to some value y, say within delta. Then I can find a neighborhood of size epsilon around some point x such that f x is within delta of y for all x in the neighborhood.
That's not quite rigorously stated, but that's the basic idea. But it's easy enough to find functions where the limit exists on both sides of some value x, but is not the same. Minor semantic note on the topic: People usually define continuity, not discontinuity.
If you asked a mathematician for a "definition of a discontinuous function", he'd be likely to say "a function which is not continuous everywhere. All excellent points. There may or may not be reasons for fine-tuning your definition of "continuous" in whatever context you're developing.
Perhaps it is of interest to the more specialized student to note that sometimes a mathematician may want to either change or extend a definition of things like "continuous" for specific purposes. The algebraic terms "field", "ring" and "group" were chosen on purpose to avoid artefact terms or labelling by humans, and to act as mnemonics. The choices had nothing to do with everyday fields, rings and groups; as teachers of algebra know, the chosen mnemonics proved useful.
Stan Brown's comparison shows that he does not know etymology of mathematical terms about which he chose to argue. The situtation of "continuous" is different: the term "countinuous" actually describes and abstracts the property "continuous".
The concept "continous" is also met by students, for the first time, at a level when the students do not yet have ability to understand the concept of a mathematical definition: at calculus courses students still think of definitions as descriptions of reality, not as agreements of concepts for embarking deductions. Thus, at that level it is pedagogically more effective to choose definitions which do not conflict with students' observations about reality.
IOW, oo! The choice of 'group' in particular was obviously related to its everyday meaning. The mathematical 'field' was not created ex nihilo; it is in fact derived from the everyday noun 'field' and therefore shares most of its etymology with that word.
But it is paedagogically very foolish to choose definitions that are in conflict with those actually used by mathematicians. And it is paedagogically valuable to have yet another opportunity to point out to students that mathematical terms have precise definitions and that they must work with these definitions, not with some vague notion carried over from everyday experience. No holes 2. When the x values are "close enough," the y-values also are close.
You can "trace" the curve without lifting your pencil. Of course, all this is very rough. Intuition is not static but developing. When one "plays" with more sophisticated examples, one comes to understand more subtleties of definition, including functions, defined on the whole real line, that are nowhere differentiable yet continuous.
If a fertile definition is awkward for the uninitiated that's unfortunate and also just tough cookies. Students must understand that what they know or deem credible is not a criterion for learning. Rather it is what professionals choose as a right path. Otherwise, one is just pandering to them. Or perhaps it is also true to say that tan x is piecewise continuous. That it only has a piecewise definition Not a mathematical question. I will not confuse either with a rope, however.
Elementary error: besides confusing a technical mathematical question with a question about the real world, you are assuming that an everyday word keeps or at least should keep exactly its everyday connotations when used as a precisely defined technical term in mathematics. Yes, but they are not labels carrying names of human beings, like in Newton's law, Euler's formula, Lagrange's theorem. This point was discussed by mathematicians, who chose the the labels 'field', 'ring', 'group'.
Related yes, in the sense that a sociological group has a leader, corresponding to the neutral element of a group, but the math concept of group does not try to abstract or develop a theory or a model for a sociological groups. In this sense, the concept of group is in sharp contrast to the concept of continuous, which abstracts everyday phenomenom of countinuity.
While you pretend to know things, can you point out what 'human being' was the one who 'derived' the label 'field'? In French the concpet of 'field' is labelled 'corps commutative'. Maybe you, as somebody who knows these things, can tell us how the the concept of 'non-commutative field' became labelled by the term 'corps' in French, and who 'derived' it?
Have you actually taught mathematics? Calculus I, II? If the students do not have appropriate cognition to assimilate a new concept to their existing cognitive structures, then students are not prepared to learn the concept. Can you give me an example of an elementary function, discussed in the calculus courses, which is not continuous, according to your definition of continuity? If no function is 'not countinuous', then 'continuous' is a concept introduced for no purpose, for the calculus courses.
I hardly had anything so far-fetched in mind. The relationship is much simpler: 'group' refers to a collection of objects. Had a completely arbitrary label really been desired, the mathematical object could have been called a 'grillip' complete neologism or a 'red' completely unrelated term, not even the right part of speech.
I have no idea. It's also irrelevant to the choice of 'field' in English. Moreover, one of the senses of 'corps' is 'collection', so it is clearly appropriate in precisely the same way that 'group' is appropriate.
Can you give me an example. Every elementary calculus text has examples of discontinuous functions. Some of them, like the floor greatest integer function arise naturally. Others are introduced to illustrate the concept of continuity, e. That is not far-fetched: to distinguish from sets of people who do not have a leader , the label 'group' was chosen to emphasize the existence of a particulat element, the neurtal element, in a group.
Thus, there is more structure in a 'corps' than in a 'groupe', which in turn has more structure than 'ensemble'. This step-wise addition of structure was the basis of choosing the names 'fields', 'group' and 'sets', which act as mnemonics for additional mathematical structures.
Step function does arise natrually in applications of mathematics, but it has not appeared in math courses prior to its introduction in conjunction with illustrations of 'continouity'.
Referring to discountinuity might be misleading for students; better say that the graph of tan x is not connected. But using the other common definition of continuity, left and right limits exist and equal the value at the point, is not a bad choice pedagogically: you have the advantage of changing your definition at more advanced courses, and thus drilling students in the practice of choosing an appropriate definition for the particular purpose at hand.
Rational functions are continuous where they are defined. Ditto for trig functions, exponentials, logarithms, and roots, sums, differences, products, quotients, and compositions, etc.
Now students can practice finding natural domains of such functions f in order to determine intervals where f is continuous and leave it at that. I have read over this thread and have the opinion that what started as a simple, legitimate question has turned into a debate of what the absolute technical definition of "continuous function" should be.
I think what is important here, especially if this is a CAL I question, is to thouroughly understand the following. I am not an expert by any means, but the definitions of "continuity" I have been exposed to talk about only two types: a continuity at a point, and b continuity on an interval, where: 1.
Although I have not formally seen "continuous function" defined, we used it informally but very frequently to describe a function that was continuous on its entire domain not necessarily on -oo,oo. Don't let technical jargon get in the way of understanding the concept of continuity. The jargon is important, but don't focus on it too much.
Seems to me those that say tan x is not a "continuous function" are focusing on an insignificant technicality that the term is equivalent to "continuous everywhere. But, it's alsa true that a dog is not a black cat. Of course, it's really of no use to say a dog is not a black cat, but you can say that and be technically correct. It seems even more sensible to call this a continuous function, even though its domain is not R. Please do not confuse "continuous everywhere" with a function that is continuous on it's entire domain, a.
Scott" wrote:. I don't believe it for a moment, if only because in English 'group' does not connote the existence of a leader. If this is in fact the reason for the choices, then you have refuted your own claim that the choices had nothing to do with the everyday meanings of the words. This is not generally true: I've seen quite a few pre-calculus texts that discuss it.
There's nothing wrong with an artificial example if it makes a point, though in most cases more natural ones could be found. Characteristic functions of simple sets are quite useful in this regard. Better yet to say what I already said in a and b above. Whatever is the reason if we go into the definition of a function and stick to the point that we are not allowed to talk of any points which are not in the domain then i guess that the problem is solved.
Cheers Ayan. In sociology and psychology of groups such connotation is given, and also in economics of companies under one holding company. But, it is irrelevant what common English says here, because that was the way the mathematicians, who chose the words 'ensemble', groupe', and 'corps' justified their choice as a mnemonics.
No, and here you are for the second time after 'labelling by humans' either misinterpreting or distorting what I wrote. I did not extend this discussion from 'continuous' to 'field'; that extension was made by somebody else; you engaged the debate and show off poor knowledge of etymology of the term 'field', which you chose to argue.
When there are teachers like you, who say that a 'continuous' function has 'points of discontinuity', no wonder why there are former students, like Jeremy Boden, who are still confused, after years of receiving their education, about 'continuity' of tan at its 'points of discontinuity'.
Technical usage is not common usage. You are talking about technical usages and, I might add, ones that may just be younger than the mathematical term. I believe that the words were chosen with some eye to their common meanings; I already said as much. I have no idea whether they were given the elaborate justifications that you imply; it's possible, but your assertion isn't evidence.
You contradict yourself. If they were chosen as mnemonics on the basis of their everyday meanings, it is obviously not true that they had nothing to do with these meanings. I did not. I pointed out that you apparently don't understand the meaning of 'etymology'. It is not limited to the reasons possibly given by someone for choosing the term; it includes the history of the word that was borrowed into technical use.
You're being dishonest. I recommend the correct explanation in a and b above, which says nothing about the function having a point of discontinuity. If you don't understand this, you've no business teaching.
Your attitude is better suited to your primary hobby of finding insignificant errors in published proofs. I will not respond again: I see no reason to give you more opportunities to 'justify' insults by selective quotation. Scott Sent via Deja. Learn what you don't. Any time "infinity" or "infinite" occurs, it's a shorthand for another meaning, that is, the statement has a different meaning than if a number were used.
In my opinion, this is a very bad approach: it is a matter of fact that discontinuous should not be read as the negative of continuous. The domain of definition makes a difference, and the most useful idea is that of continuous extension.
Almost any mathematician would say that the tangent function is continuous inside its own domain of definition. Definition just do not cover such a situation. You may also want to consult the Definition of continuity or Classification of discontinuities.
As others have mentioned, considering continuity outside domain doesn't make sense. The extension makes full "circles" in a continuous way, and a restriction of a continuous function is again continuous. In my opinion, the definition of continuity from your textbook is non-standard when applied to functions defined on sets with isolated points. One way to interpret point 1. Sign up to join this community. The best answers are voted up and rise to the top.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Ask Question. Asked 9 years, 3 months ago. Active 3 years, 6 months ago. Viewed 18k times. Anurag Kalia Anurag Kalia 1 1 gold badge 3 3 silver badges 13 13 bronze badges. Add a comment. Active Oldest Votes. Siminore Siminore Konrad Sakowski Konrad Sakowski 3 3 silver badges 13 13 bronze badges. They are dead silent when talking about discontinuity, whether f c needs to be defined or not adding to the confusion.
How can they be same? There is at least a minus sign as a point of difference between the two. Yet real line is infinite in length! Do we treat the radius of this circle to be infinite too?
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