Continuity of Composite Functions + Examples

in MES Science2 years ago

In this video I go over a simple theorem on the continuity of composite function which when simply stated says that a continuous function of a continuous function is a continuous function. I also go over a few useful examples on the continuity of composite functions.


Watch video on:

Download video notes: https://1drv.ms/b/s!As32ynv0LoaIiMdLp1-2vb7B1guw6A?e=bqxsZa


View Video Notes Below!


Download these notes: Link is in video description.
View these notes as an article: https://peakd.com/@mes
Subscribe via email: http://mes.fm/subscribe
Donate! :) https://mes.fm/donate
Buy MES merchandise! https://mes.fm/store

Reuse of my videos:

  • Feel free to make use of / re-upload / monetize my videos as long as you provide a link to the original video.

Fight back against censorship:

  • Bookmark sites/channels/accounts and check periodically
  • Remember to always archive website pages in case they get deleted/changed.

Buy "Where Did The Towers Go?" by Dr. Judy Wood: https://mes.fm/judywoodbook
Subscribe to MES Truth: https://mes.fm/truth

Join my forums!

Follow along my epic video series:


NOTE #1: If you don't have time to watch this whole video:

Browser extension recommendations:


Continuity of Composite Functions

Continuity of Composite Functions.jpg

Theorem

If g is continuous at a and f is continuous at g(a), then the composite function (f◦g)(x) = f(g(x)) is continuous at a.

“a continuous function of a continuous function is a continuous function”

Proof

Since g is continuous at a, we have:

image.png

Since f is continuous at g(a), we can apply the Limit of a Composite Function Theorem (see my earlier video) to obtain:

image.png

Where are the following functions continuous?

image.png

image.png

image.png

image.png

image.png

Sort: