Why is continuity of a function important

Therefore, it is only natural to have a word for this property and to properly define it to ensure that everybody is talking about the same thing. Moreover, it is worth studying whether any useful properties follow from continuity — which is indeed the case — as those properties would have a broad application. Without having a concept of continuity, we would have to appeal to intuition, make assumptions or explicitly show the fact, whenever we would normally apply the intermediate-value theorem or swap limits and function evaluation.

Moreover, only by knowing what we usually take as given due to the ubiquity of continuity, we know what we cannot expect anymore in the rare case that continuity is absent. The above motivation translates to almost all of abstract mathematics for most people and most non-mathematicians: We detect and abstract some properties of natural objects and see whether we can derive some new abstract statements from those properties, which we can in turn translate back to application, where they may be useful.

In this sense the motivation to study continuity is not much different than the motivation to study, e. I agree with user BenCrowell that continuity is less essential in freshman calculus than other topics such as differentiability. Perhaps it is for this reason that Keisler treats differentiability first in his online textbook which we have used for the past three years to teach freshman calculus to a total of almost students by now.

To address the OP's question specifically, I would therefore suggest motivating differentiability first, which is an easier task since it suffices to mention applications in physics velocity, etc and other fields. Then one could point out that a useful larger class includes continuous functions which are useful for technical reasons, such as for example wanting to work with the absolute value function or the cube root function.

A possible objection that continuity is more general and therefore should be treated first is not convincing; after all we don't start freshman calculus with Lebesgue integrable functions. As others, I think it is not necessarily relevant to actually study continuity.

I do mention it in order not to lie when I have to e. However, depending on the context, there are reasons one can chose to study it in details. The most important I can see is that proving things around continuity is a very good model of what mathematics, especially analysis, looks like. For example, proving that the product of two continuous functions is continuous gives already gives a rather sophisticated proof for freschmen.

The main question I would ask myself if I had to teach a course where I could, or could not treat continuity in some depth, is whether I want to show my students such proofs I could answer positively even for non-math majors , and whether I want to ask them to be able to perform them this would probably be restricted to math majors.

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. Why should we study continuity? Ask Question. Asked 5 years, 7 months ago. Active 5 years, 7 months ago. Viewed 4k times. My attempt to answer the question: I would argue that there are two main reasons to study continuity: Continuous functions have "beautiful" properties like the intermediate value property for connected domains , the limit can be put inside the function i.

In topology continuous functions are exactly those functions which preserve topological structures. Did I miss something? How would you argue why continuity is important for mathematics? Improve this question. Community Bot 1. Stephan Kulla Stephan Kulla 1, 8 8 silver badges 26 26 bronze badges. For the vast majority of students taking that type of course, it would probably be fine if the treatment of continuity was half a page in the text and 10 minutes in lecture.

Very useful information, we IB teacher required ,this type of information for all the concepts Mathematics. Continuity of functions Learn Maths Online. Continuity of functions The word continuous means without any break or gap. Reasons for dis Continuity of functions i does not exist i.

Removable type of discontinuity can be further classified as: a Missing Point Discontinuity- Where f x exists finitely but f a is not defined. Non-Removable type of dis Continuity of functions In case f x does not exist then it is not possible to make the function continuous by redefining it. Thank you very much, sir please suggest how can we make it more useful for our students?

Si, please sand 11th or 12th maths paper level of iit advance and maind. Okay, as the previous example has shown, the Intermediate Value Theorem will not always be able to tell us what we want to know. So, remember that the Intermediate Value Theorem will only verify that a function will take on a given value.

It will never exclude a value from being taken by the function. Also, if we can use the Intermediate Value Theorem to verify that a function will take on a value it never tells us how many times the function will take on the value, it only tells us that it does take the value. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode.

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Example 2 Determine where the function below is not continuous. Example 3 Evaluate the following limit.