1. Read the post below
2. Watch the video linked below
3. Work on the worksheet for the remainder of class
1. Read and take notes please
We learned last class about continuity. There are three types of discontinuities:
- Removable: the roads meet, but there either is no bridge, or the bridge is in the wrong place. (mathematically: limit exists)
- Jump: the roads do not meet each other. Bridge is irrelevant here. (math: left limit does not equal right limit, but both are finite)
- Infinite: one or both of the roads never meets the river (vertical asymptote)
There are some subtleties to this:
- Functions are continuous except for where they are not. So for example in the piecewise cases you'll see below there is no need to worry about, for example, any x-value except for 3. Because x^2 is always continuous and 2x-4 is always continuous. What matters is if the two of them are continuous together (namely, at x=3 in the first example)
- Piecewise functions are usually jump discontinuities. Take a limit where the two pieces meet, and if you get two different finite numbers from either side, then it's a jump. See below:
- Sometimes piecewise functions are removable discontinuites though. Consider this: