Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

5: Continuity

( \newcommand{\kernel}{\mathrm{null}\,}\)

”Continuous learning is the minimum requirement for success in any field.” - Brian Tracy

We now have a wide variety of functions at our disposal, and the ability to take limits of them. Chapter five will introduce us to the definition of continuity as well as discuss what being a continuous or discontinuous function will look like when graphed.

We say that functions that agree with their limit at a given point are continuous. We start off by defining the four different types of discontinuities: removable, infinite, jump, and oscillating. This allows us to appreciate fully what it means for a function to be continuous.

This naturally leads us to discuss the only exciting continuity examples: piecewise defined functions. These Frankenstein functions are made by putting together different parts of different functions. These will allow us to blend together some of the previous functions we have discovered, and still use limits to understand them.

We have a brief review for solving systems of equations, and are introduced to the most famous piecewise function: the absolute value function. We ensure we can solve equations and inequalities using the absolute value, as well as take limits of functions which use them as a component.

We finish out with our first theorem in the book, the Intermediate Value Theorem. This helps us understand why being a continuous function is important, and lets us know if a solution exists to an equation in certain situations.


This page titled 5: Continuity is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Ken Huber.

Support Center

How can we help?