Continuity

Karl Weierstrass (1815 – 1897) was the mathematician who (finally) formalized the definition of continuity. Included in that definition was the epsilon-delta definition of limit. This definition has been pulled out, so to speak, and now is usually presented on its own. So, which came first – continuity or limit? The ideas and situations that…

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…

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…