If the topic of continuity of functions is explained theoretically and by solving some examples numerically, students face difficulties in understanding the issue of continuity of functions. In my opinion, the best way to learn continuity of functions is to use engineering applications. Through shapes, the concept of continuity of functions deepens in students' minds.

Do you have any specific functions in mind, Maged?

Many undergraduate students find difficulties in learning the concept of continuity, as it look like a tough mathematical theory. In my view, the major difficulty they are facing is they are not able to visualize the concept graphically. Before introducing the theoretical notion of continuity to undergraduate students, educators should provide examples using graphs in a simple language. Then explain each and every term in the definition graphically. This may leads to better understanding of continuity.

I think that the previous answers pertain mainly to students who are beginners in calculus; i.e. mainly to pre-college students. There also exist the students who are familiar with calculus but beginners in analysis, like most undergraduates. Those students have a working knowledge of the basic tools of mathematical analysis and therefore they have acquired an intuitive understanding of continuity, but they may not be familiar with epsilon-delta definitions. For those students, the challenge regarding the notion of continuity of a real function is to understand, digest and, mainly, apply to proofs the epsilon-delta definition.

One can find various published articles that analyze and propose solutions regarding the difficulties that students face in understanding epsilon-delta definitions like that of continuity. However, a point that is not properly mentioned and I think it is useful is the analysis of the statement of the epsilon-delta definition of continuity as a predicate-logic formula. Such an analysis would help students to understand the logical structure of the definition of continuity and also to learn how to correctly negate multiply quantified statements, which is essential in understanding the proofs of many theorems in analysis.

Another point that I think merits attention regarding continuity is the relation between the epsilon-delta definition and the so-called Heine definition, which in literature is referred to as the sequential criterion for continuity. The epsilon-delta definition is a stronger statement than the Heine definition, since the former implies the latter and for the latter to imply the former, the Axiom of Choice must be invoked. This point is not mentioned in literature, when the epsilon delta definition is proved by the Heine definition, and most students ignore this important point that, if anything else, indicates the connection of all those definitions to the axiomatic set theory, which is a foundational part of modern mathematics.

Prof. Naimpally and several others suggested nearness as the most

convenient mean.

One can use the nearness of two sets or two nets, and their particular

instances.

For instance, nearness between two points, between a point and a set,

and two sets.

Nearnessess (closures) can be most nicely derived from relations and

relators (families of relations).

Prof. Kuratowski and others called a function f to be continuous if

x\in \cl(A) implies f(x)\in \cl(f[A]).

That is, if the point x is near to the set A, then the point f(x) is near to

the set f[A].

This can be written in the shorter form that f[\cl(A)]\subseteq \cl(f[A]),

or even f o \cl\subseteq \cl o f.

This already indicates that continuity properties are actually

inclusion properties for compositions .

In this sense, we have suggested several continuity properties not only

of functions and relations, but also those of relators.

However, leading topologists usually rejected our papers with great

pleasure. Dishonesty is continuously increasing not only in politics.

Hello Margaret MacDougall ,

Just a word or two of praise for your wonderful question. As a student, I was always bemused by proofs in calculus, especially those having to do with continuity. The proofs seemed to appear from the heavens like manna, or, in my case, anti-manna since they were so indigestible, intellectually speaking. When the professor would ask the class if there were any questions with regard to the proofs, I was too cowardly to raise my hand, and even if I had had the chutzpah to signal my displeasure, I was so clueless that my posed question would have been incomprehensible to everyone including myself. In short, I felt myself to be a child of a lesser god.

Many years later in another mathematics course, the professor let us in on the secret. Simply put, he stated that when a proof was submitted to a journal, all the scaffolding used to generate the proof was carefully removed, and only a few steps along with the conclusion were presented in the finished paper. The book about Paul Erdős [1] reinforced this point when it mentioned, on p. 241, that even Erdős admitted later in his life that he could not follow the proofs in his own papers written some 30 or 40 years earlier. Even if a published proof is in error, mathematicians can still be obtuse in explaining the error, see [2], where Ian Stewart relates the story about Leslie Lamport's discovery of an incorrect proof of the Schroeder-Bernstein theorem, and the issues Stewart had in verifing the error from the few breadcrumbs left by Lamport.

I know I have taken up too much of your time, already, but let me just mention one other reason why proofs are so incomprehensible. From the time Isaac Newton (1642-1727) codiscovered fluxions (differential calculus) circa 1665 with Gottfried Wilhelm Leibniz (1646-1716), the foundations (continuity and differentiability) of this new discipline were in doubt. Bishop George Berkeley (1685-1753) in 1734 was perhaps the first person to point out that while fluxions was useful and, mostly likely correct, its foundations were wanting. In fact, it was not until the 1820s that the books by Augustin-Louis Cauchy (1789-1857) placed the foundations of differential calculus firmly in the modern era, though that is not to say that they (the proofs) were anymore popular with students. How easy can the proofs of continuity and differentiability be if it took some of the best mathematicians more than a century to establish the epsilon-delta proofs, we use today, see [3-4]?

[1] Paul Hoffman; The Man Who Loved Only Numbers; Hyperion; 1998; p. 241.

[2] Ian Stewart; Secret Narratives of Mathematics; in John L. Casti, Anders Karlqvist (editors); Mission to Abisko; Helix Books (Perseus Books); 1999; pp. 157-160.

[3] Carl B. Boyer; A History of Mathematics; Princeton University Press; 1968. pp. 469-470 & 561-564.

[4] Morris Kline; Mathematics, The Loss of Certainty; Fall River; 1980; pp. 192-194.

Regards,

Tom Cuff

Hi professors. I hope you are doing well. This is a mathematical article that proves a new recurrence relation that is fundamental for mathematics. The article proves also that four infinite series are equivalent. Hence, this article opens new opportunities to demonstrate and develop new mathematical findings and observations. This is the link: https://www.researchgate.net/publication/364651911_A_useful_new_equation_of_four_infinite_series_and_sums_by_using_a_new_demonstrated_recurrence_relation

One may teach the derivatives of discontinuous functions, allowing algorithms for quantum computation. The concept of "continuity" is not physical, although mathematically invented. See "Quickest Calculus", available in paper or as a free PDF.

If a student gets clear idea about Limit of a real-valued function at a point, then they can easily understand the notion of continuity. Teachers face challenges to make the concept of Limit easy to students.

Continuity simply means not to lift the pen while drawing, but all kinds of sharp angles could be included.

Analytic functions

F(x) = a0 + a1 x + a2 xx +... finite or convergent may be better behaved.

Then there are several ways to get derivatives.

Dear Juan Weisz, you have told that 'Continuity simply means not to lift the pen while drawing, but all kinds of sharp angles could be included.'

Then what's about the continuity of the real valued function

\[

f(x) = \dfrac{1}{x}

\]

at $x = 0$? It is continuous its domain (having a discontinuity at $x = 0$). But the graph of this function can not be drawn without lifting the pencil from the page.

So I think that every function graph of which can be drawn without lifting the pencil from page is continuous, but the converse is not always is not true.

Yes, obviously divergences count against continuity. You are correct.

MM: Mathematical real-numbers are non computable and imprecise, being not physical. Comtinuity does not exist in nature.

See examples of functions that are everywhere not continuous, and yet have differentials, and details at

Article Algorithms for Quantum Computation: The Derivatives of Disco...

The number 3 can be divided exactly by 3, in base 10, using modular arithmetic with no infinitely long numbers. 0 mod 3 = 0; 1 mod 3 = 1; 2 mod 3 =2; 3 mod 3 = 0. INTEL uses tri-state in more efficient digital circuits, where the third-state is Z, representing high-impedance.

Tri-state is so popular it is a trademark, owned by a digital chip company in the U.S., and covered in a book by Mano, which details Verilog as its coding paradigm, which is a standard by IEEE.

Traffic lights use 3 states. The U.S. constitution enshrined 4 powers, including the press.

Life, math, and physics are more complex than Boolean algebra. The photon follows at least 3 states in interactions with matter, as known since 1917. This deprecates qubits.

Information is not a neutral fluid, that could be only blocked or let-pass, as modeled by a relay. There is also, since the year 2,000, information encoding -- used in MITM attacks and for speed with higher cybersecurity. This deprecates bits, and Shannon theory.