Which Came First?

In one of my math classes – it may have been calculus – many decades ago, we started by determining what kind of functions we were going to study. A good part of the answer was continuous functions. Looking closely, you will find that almost all the theorems in beginning calculus require that the function be continuous on an interval as one of their hypotheses (The interval could be all Real numbers.) Later theorems require that the function be differentiable, but, as you will learn, if a function is differentiable, then it is continuous. So, calculus studies continuous functions (or those that are not continuous at only a few points).

A function is continuous on an interval, roughly speaking, you can draw its graph from one side of the interval to the other without taking the pencil off the paper. Thus, if a function has a hole, a vertical asymptote, a jump, or oscillates wildly it is not continuous. Continuity is first determined for a function at a point in it domain. Then this is extended to all the points in an interval.

Students come across functions that are not continuous long before they encounter calculus and limits. They see functions with asymptotes, jumps, and holes long before calculus. Discussing continuity gives a reason to talk about limits informally and how the idea of “getting closer to” a point works. This eventually leads to the idea of a limit and the need to define the term.

The definition of continuity at a point that is used most often is this:

A function f is continuous at x=a if, and only if, (1)  f\left( a \right) exists, (2)  \underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right) exists, and (3)  \underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right)=f\left( a \right).

The first two conditions are probably included to prevent beginning students from thinking that if the value and the limit are both “infinite” as in the case with some vertical asymptotes, then the function is continuous. In fact, the two things can only be equal if they are finite.

The definition of limit (which is not tested on either AP Calculus exam) states that

 \underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right)=L, if, and only if, for every number \varepsilon >0 there exists a number  \delta >0 such that if \left| {x-a} \right|<\delta , then \left| {f\left( x \right)-L} \right|<\varepsilon .

It was in the early 1800’s that the epsilon-delta definition of limit was first given by Bolzano (whose work was overlooked) and then by Cauchy, Jordan, and Weierstrass..Historically, the definition of continuity was first given by Karl Weierstrass (1815 – 1897) and  Camille Jordan (1838 – 1922). Their definition is:

A Real valued function is continuous at  x=a, if and only if, for every number \varepsilon >0 there exists a number  \delta >0 such that if \left| {x-a} \right|<\delta , then \left| {f\left( x \right)-L} \right|<\varepsilon .

As you can see, the original definition is simply the modern definition of limit applied to the concept of continuity.

So, which came first continuity or limits?

Calculus textbooks and the 2019 Course and Exam Description for AP Calculus’s first unit begins with limits (lessons 1.2 to 1.9) and then continuity (lessons 1.10 – 1.16). They are being logical: the concept of limit is needed to define continuity.

So, logically you need limits to talk about continuity. Practically, continuity, or lack thereof, comes first. Students should be familiar with continuous graphs and the types of discontinuities before they start calculus. The calculus course will formalize things and make the ideas precise using limits.

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Stretch your brain a bit: Almost all the functions you will study are continuous at all but a few (a finite number) of places. If that were not so, there would not be much calculus you could “do.” But, consider the Dirichlet function:

D\left( x \right)=\left\{ {\begin{array}{*{20}{c}} 0 & {\text{if }x\text{ is irrational}} \\ 1 & {\text{if }x\text{ is rational}} \end{array}} \right.

Since there are always rational numbers between any two irrational numbers, and irrational numbers between any to rational numbers, this function is not continuous anywhere! No point is adjacent to any other point.

And a little more stretch: Discuss the continuity at x = 1 of this function:

L\left( x \right)=\left\{ {\begin{array}{*{20}{c}} x & {\text{ if }x\text{ is irrational}} \\ 1 & {\text{if }x\text{ is rational}} \end{array}} \right.


Next Tuesday, I will begin posting the lists of references to blog posts about topics related to the units of the  2019 Course and Exam Description for AP Calculus beginning with Unit 1: Limits and Continuity.


 

 

 

 


 

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