Section 5.2 Continuity: Your sentence motivating the idea of continuity

[markbarsamian]

Section 5.2 Continuity: Your sentence motivating the idea of continuity is problematic in a number of ways.

To start with, you say

"...plugging the value into the function."

This ought to say more clearly

"...x-value...."

rather than just

"...value...."

But more importantly, at this point in the text, students have not computed any limits by using the formula for a function. They have only found limits by looking at graphs. So your motivation for the concept of continuity really ought to refer to graphs. Here is the way that I motivate it in my course. (My introduction has elements of the "lift the pencil" idea that you refer to later, when discussing continuity on an interval.)

Introduction to Limits that I use in my course

Observe that we have seen some behavior of graphs that is obviously "weird"

Hole in graph at x = a, with missing dot
Hole in graph at x = a, with dot in wrong place
Jump in graph at x = a.

A unifying informal description of all of these could be that one must lift the pencil to draw the graph in the vicinity of x = a. (One might think that one does not have to lift the pencil if there is hole in the graph, but rather can just go around the hole like a roundabout in a road and then proceed to the rest of the graph. But in fact, the open circle at the location (x,y) = (a,L) is meant to make it clear that there is no dot at that location. So strictly speaking, one would draw the graph by approaching x=a from the left, lifting the pencil over the location (x,y) = (a,L), and then putting it down to the right of that location and proceeding with the rest of the graph.)

On the other hand, speaking informally, much of the graph can be drawn without having to lift the pencil from the page.

Is there some more precise, more concise, way of articulating the idea of being able to draw the graph in the vicinity of x = a without having to lift the pencil?

There is, using the terminology of limits and function values. The concept is given a name: "continuity".

Definition: To say that a function is continuous at x = a means that the following three things are true
(1) f(a) is defined. That is, there is a point on the graph of f(x) at the location (x,y) = (a,f(a))
(2) The limit of f(x), as x approaches a, exists. That is, the graph appears to be heading for some location (x,y) = (a,L)
(3) The limit of f(x), as x approaches a, equals f(a). That is, the location (x,y) = (a,L) that the graph appears to be heading for is the same location (x,y) = (a,f(a)) where there is a point on the graph.

Informally, one could say that a function is continuous at x = a if the graph of f(x) can be drawn near x = a without lifting the pencil.
******************** End of introduction to Limits that I use in my course *****************