1. Ch. 2 – Limits and Continuity

2. Therefore, f(x) is discontinuous at x = 1 and x = 5 over [0, 5].
A function f(x) is continuous if for all values of c on a specified interval.
Basically, the graph can’t have any holes, asymptotes, or breaks!
For an interval [a, b], the endpoints are defined as continuous if
Ex: Find the points of discontinuity of the function graphed below over [0, 5].
Therefore, f(x) is discontinuous at x = 1 and x = 5 over [0, 5].
Get used to stating the interval in your answer!

3. Types of Discontinuities (all at x = 2)
Removable Discontinuities
Jump Discontinuity
Infinite Discontinuity
Oscillating Discontinuity

4. Ex: Find the points of discontinuity of the following functions and state the type of discontinuity.
f(x) has an infinite discontinuity at x = 0 over the real numbers.
f(x) has a removable discontinuity at x = -1 over the real numbers.
f(x) is discontinuous outside its domain; that is, from (-∞,-1) and (1, ∞)

7. Extended Functions
Ex: Write an extended function for f(x) that is continuous for all real values of x.
Extended function = piecewise function with f(x) and value for all of the removable discontinuities
This function has a hole at x = 1, so let’s fill in the hole…
Since f(x) approaches -1 at x = 1, our answer is as follows:

8. Sketch a possible graph.

9. Sketch a possible graph.