Continuity

Karl Weierstrass (1815 – 1897) was the mathematician who (finally) formalized the definition of continuity. In that definition was the definition of limit. So, which came first – continuity or limit? The ideas and situations that required continuity could only be formalized with the concept of limit. So, looking at functions that are and are…

Limits

In an ideal world, I would like to have all students study limits in their precalculus course and know all about them when they get to calculus. Certainly, this would be better than teaching how to calculate derivatives in precalculus (after all derivatives are calculus, not pre… ). Here are a few of my previous…

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…

Determining the Indeterminate

When determining the limit of an expression the first step is to substitute the value the independent variable approaches into the expression. If a real number results, you are all set; that number is the limit. When you do not get a real number often expression is one of the several indeterminate forms listed below:…