Section 1.5 Continuity
Module Learning Objectives.
- Define what it means for a function to be continuous and classify different types of discontinuities.
- Decide where a piecewise defined function is continuous.
- State the Intermediate Value Theorem and describe the intuition behind.
Video 1: Continuous Functions and Four Types of Discontinuities.
This video introduces the intuitive notion of continuity, the precise definition, and describes four types of discontinuities.
Post-video reflection.
Think about all the common functions you can name, such as \(tan(x)\text{.}\) Where are each of them continuous?
Video 2: Continuity of Piecewise Functions.
When our functions are piecewise defined, we need to be particularly careful about checking continuity at the point where the definition changes.
Post-video reflection.
Suppose the definition of the piecewise defined function changes at a values \(x=a\text{.}\) Is \(a\) the only spot you need to check for continuity, or do you need to still check the rest?
Video 3: Intermediate Value Theorem.
This video is all about the Intermediate Value Theorem.
Post-video reflection.
- We use IVT (without thinking about it) all the time in our lives, just like with the claim “you were once 3 feet tall.” What is another everyday statement you might say to someone who doesn’t know calculus but is implicitly using IVT?
- We didn’t prove IVT – and we won’t in class either. The theorem depends on a key property of the real numbers called “completeness” which in essence is about the idea that the real numbers don’t have any holes. We’ll talk about this a lot if you take Math 236 Real Analysis in the future!
