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This question is related to How can I motivate the formal definition of continuity? Imagine a student asks the question why it is worth it to study continuity. What is a good response to this question?

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.e. $\lim_{n\to\infty} f(x_n) = f(\lim_{n\to\infty} x_n)$), compositions like sums and products of continuous functions are again continuous etc.
  • In topology continuous functions are exactly those functions which preserve topological structures.

However, the second reason can only be fully understand when the student already has studied topology.

Did I miss something? How would you argue why continuity is important for mathematics?

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It's not obvious to me that continuity should be studied in freshman calculus. 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. Certainly it's important for students who are math majors, but those are a tiny fraction of all students in freshman calc. The underlying problem is that there is only one flavor of freshman calc, which is expected to serve many different kinds of students. What are biology majors gaining from a detailed discussion of continuity? – Ben Crowell Mar 20 at 15:55
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I generally think that the word "beautiful" is overused in cases like this. I'd prefer a descriptor like useful, convenient, powerful, tractable, etc. – Daniel R. Collins Mar 20 at 18:56
    
it is important to understand the intuitive idea of continuity in part to draw attention to the vast contrast of the discrete. The contrast between discrete and continuous variables is something which both mathematicians and applied students of the mathematical sciences must both be aware. – James S. Cook Mar 22 at 18:00

Most functions that are studied by physicists and other scientists are continuous. However, more and more discontinuous functions are appearing in the various sciences. This is due to:

  • Computers and their digitization of data. Many computer routines produce discontinuous output, even if the data is near-continuous.
  • Quantum theory is a mixture of the continuous and the discontinuous. Schrodinger's equation assumes continuity and differentiability, but in the Copenhagen interpretation of quantum theory any measurement causes a collapse of the waveform and things become discontinuous: the spin is either up or down, etc.
  • The study of Chaos, resulting from weather and other such topics, gives results that are practically discontinuous, even if continuous in theory.
  • Catastrophe theory, developed in the 1970's, shows there are a variety of ways to get discontinuity, with a variety of applications.

Any student who does not directly think about continuous and discontinuous functions early in his/her math training may assume that all functions are continuous. This student will get into trouble when this is not the case.

Example story: when I begin to teach the Intermediate Value Theorem for continuous functions, I give the example of me driving to school from my home. Students assume that the distance from my home is a continuous function, so I must at some point be exactly ten miles from my home (school is 22 miles away). This assumption is correct, but when I ask them they cannot even imagine any discontinuous possibility such as a Star Trek beam transporter. This silly example shows me every year that my students do not understand how some functions can act differently from others. They have to be shown the differences between continuity and the lack of it.

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Imagine a student asks the question why it is worth it to study continuity. What is a good response to this question?

  1. It matches human intuition. We use continuity very frequently without realizing we're using it. It's the foundation of most approximations we make in physical science. Whenever we think $f(x)$ can be approximated by $f(y)$ because $x$ can be approximated by $y$ we inherently assume $f$ is continuous.

  2. It provides a natural way of understanding irrational exponents (or more generally, extend a function defined on rational number to real number). How do you define $3^{\sqrt2}$ or $7^{\pi}$? When we're first introduced exponential, we're taught $a^b$ means "$a$ multiply $b$ times" when $b$ is an integer. When b is a rational number, like $5^{1/2}$, it's also not too hard to think it's a positive number that "get 5 by multiplying itself". However it's very difficult to explain what $7^{\pi}$ means naturally. To me, the easiest way to understand it is "that is a number where this sequence $7^3, 7^{3.1}, 7^{3.14}, 7^{3.141}, 7^{3.1415}......$ eventually get closed to." More generally and rigorously, we use the following corollary from continuous function,

Continuous functions preserve limit of sequence. Any real number is a limit of a sequence of rational numbers. Therefore, if $f$ is a real value continuous function and we know it's value on each rational number, we know $f$. In other words, continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ can be uniquely determined by it's value on $\mathbb{Q}$.

IMO it's a much more natural way of understanding exponential than the "power series way" or "inverse function of antiderivative of $\frac{1}{x}$ way" many calculus textbooks use. That kinds of attempts to avoid "completeness of $\mathbb{R}$" are unintuitive and often create confusion to high school/freshman college students.

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In our intuition, everything is continuous. But if we think long enough about real life, we discover that actually lots of things aren't. Computers generally replace continuous things with discontinuous approximations. When a driver hits the car's brake, acceleration is discontinuous. The stock market has lots of discontinuous jumps (ignore that at your peril).

Much of the mathematical study of continuity was developed in trying to understand why something unexpected happened when in fact things weren't continuous. When a function is continuous, a large set of tools is available to us to use. When it is not continuous, these same tools will lead us astray. It is important for us to understand when we are and aren't allowed to use these tools. To understand it, we need to understand what continuity means.

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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 300 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.

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Indeed, in many ways I'd say I don't care much about continuous-but-not-differentiable functions... since we can't differentiate them in elementary terms (nevermind distributionally, for a moment). I think it's not the case that we care about that vast ocean of non-differentiable continuous functions that Baire category tells us is all around us, though not easily identifiable individually. In practice, it's more like piecewise differentiable, with jump discontinuities and corners in the graph, etc. Yes, these are "not continuous", and that's even stronger than "not differentiable", but... – paul garrett Mar 22 at 21:42

Continuity is a property that many real functions or relations have on practically relevant scales, i.e., if there are discontinuities, they are so small that they have no impact on most applications. For example, we treat space and time as continuous, though they may be discretised on levels that we cannot yet access with measurements, and we can treat matter as continuous on many scales, if even though it is not continuos on the atomic level.

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.g., integers: Not every number occuring in real life is an integer but many are and properties of integers may thus be of general interest.

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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.g. when stating the fundamental theorem of calculus), but often barely more.

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).

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