Chapter One Section Four Continuity and One-Sided Limits

Chapter One Section Four So, what does it mean if we say that a function is continuous? We will now look at a series of slides depicting different functions. Without having defined the word continuous as it applies to functions, I want you to think about whether each graph is the graph of a continuous function. Where you think that the answer is NO, try to explain to yourself why it is not continuous.

Chapter One Section Four

So, what is a continuous function anyway? We do not formally define the word continuous in this large a region. What we talk about is continuity at a point, or in a neighborhood. Do you remember me using that word previously? What follows is a very formal definition for a word we will use all year in this course.

Chapter One Section Four A function f is said to be continuous at c if, and only if, the following three conditions are met:

Chapter One Section Four A rather natural extension of this definition is to discuss continuity as a function-wide feature. We can only say that a function is continuous if it is continuous at every point in its domain. Even then, we need to be careful so that we are clear that we are only talking about the continuity of the function within its domain. This seems like an esoteric point, but we will try to clear this up as we move along.

Chapter One Section Four However, at this point we want to be careful to always include a region, or a domain interval, in our conversation of continuity. The other part of the name of this section is the phrase one-sided limits. What might this mean?

Chapter One Section Four Examine the piecewise defined function below. Each grid mark on the graph represents one unit. Come prepared to discuss how we can talk about the limit of this function as x approaches 4.