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 required continuity could only be formalized with the concept of limit. So, looking at functions that are or are not continuous helps us understand what limits are and why we first need them.
In the ideal world, students would have plenty of work with continuous and not continuous functions before starting the calculus. The vocabulary and notation, if not the formal definitions, would be used as early as possible. Then when students got to calculus, they would know the ideas and be ready to formalize the ideas.
The Intermediate Value Theorem (IVT) is an important property of continuous functions.
Using the definition of continuity to show that a function is or is not continuous at a point is a common question of the AP exams, as is the IVT.
Continuity The definition of continuity.
Continuity Should continuity come before limits?
From One Side or the Other One-sided limits and one-sided differentiability
How to Tell Your Asymptote from a Hole in the Graph From the technology series. Showing holes and asymptotes on a graphing calculator.
Fun with Continuity Defined everywhere and continuous nowhere. Continuous only at a single point.
Theorems The Intermediate Value Theorem (IVT) and suggestions on teaching theorems.
Intermediate Weather Using the IVT
Right Answer – Wrong Question Continuity or continuity “on its domain”?
Revised from a post of August 22, 2017