# From One Side or the Other.

Recently, a reader wrote and suggested my post on continuity would be improved if I discussed one-sided continuity. This, along with one-sided differentiability, are today’s topic. The definition of continuity requires that for a function to be continuous at a value x = a in its domain  and that both value are finite. That is, the…

# Continuous Fun

The topic of this post is continuity. The phase “a function is continuous on its domain” was much discussed last week on the AP Calculus Community bulletin board as it is about this time every year. This led to a discussion of one-sided continuity at the endpoint of an interval. Let’s start by looking at…

# Right Answer – Wrong Question

About this time every year the AP Calculus Community discussion turns to the sentence, “A function is continuous on its domain.” Functions such as  cause confusion – is it  continuous or not?  The confusion comes, I think, from the way we introduce continuity to new calculus students. We say – and I did say this…

# Continuity

The definition of continuity of a function used in most first-year calculus textbooks reads something like this: A function f is continuous at x = a if, and only if, (1) f(a) exists (the value is a finite number), (2)  exists (the limit is a finite number), and (3)  (the limit equals the value). A…

# Definitions

Definitions are similar to theorems, but are true in both directions; technically, this means that the statement and its converse are both true (). The double arrow is read “if, and only if.” Both parts are either true or both parts are false. Definitions usually name some thing or some property.  Definitions are not proved.…

# Fun with Continuity

Most functions we see in calculus are continuous everywhere or at all but a few points that can be easily identified. But consider the Dirichlet function: Since there is one (actually many) rational numbers between any two irrational number and one (many again) irrational numbers between any two rational numbers, this function is not continuous anywhere! But a…

# Continuity

Limits logically come before continuity since the definition of continuity requires using limits. But practically and historically, continuity comes first. The concept of a limit is used to explain the various kinds of discontinuities and asymptotes. Start by studying discontinuities. Types of discontinuities to consider: removable (a gap or hole in the graph), jump, infinite…