1. AP Calculus September 6, 2016 Mrs. Agnew
Continuity
AP Calculus
September 6, 2016
Mrs. Agnew

2. Essential Question Essential Vocabulary
What does it mean for a function to be
continuous?
Essential Vocabulary
Continuity
Discontinuity (jump, hole, break, gap)

3. What is Continuity?
A function that is continuous at x = c has no jumps, holes, or gaps at c.
Let’s examine some functions that are not continuous due to the following:
Hole
Jump
Gap

4. Continuity at a Point A function f is continuous at x = c if
the following conditions are met:
Continuous when
limit can be found
using method of
direct substitution.
Let’s examine a graph…

5. Continuity on an Interval
A function is continuous on the open interval (a,b) if it is continuous at every point in the interval.
A function that is continuous for all real values of x is said to be everywhere continuous.
Give some examples of functions that are everywhere continuous.

6. Discontinuities There are two types of discontinuities:
Removable (holes)
Nonremovable (breaks, gaps)
A function that can be factored and the discontinuity canceled has a removable discontinuity.
Examples

7. Continuous Functions Polynomials: continuous everywhere
Rationals: continuous over domain
Abs Value: continuous everywhere
Trig Functions
Sine and Cosine: continuous everywhere
Others: continuous over domain
Radicals: continuous over domain
Exponential: continuous everywhere
Logarithmic: continuous over domain

8. Intermediate Value Theorem
Suppose f is continuous on the open interval (a,b) and suppose that N is a number between f(a) and f(b). Then there exists a number x between a and b such that f(x) = N.

9. Notes on the IVT
The IVT is an existence theorem… tells us a value exists, not what the value is.
Usefulness:
Proving that roots of a polynomial exist
How a calculator graphs…
Examples

10. Homework: 9/6/16
Page 79 – 82
#10, 12, 15, 33, 43 – 49, 53, 54, 61 – 67 (O)
68, 83, 85, 93, 99 – 103, 108