1. Continuity and One-Sided Limits
Lesson 2.4

4. Don't let this
Happen to you!

5. Intuitive Look at Continuity
A function without breaks or jumps
The graph can be drawn without lifting the pencil 

6. Continuity at a Point A function can be discontinuous at a point
A hole in the function and the function not defined at that point
A hole in the function, but the function is defined at that point

7. Continuity at a Point A function can be discontinuous at a point
The function jumps to a different value at a point
The function goes to infinity at one or both sides of the point, known as a pole

8. Definition of Continuity at a Point
A function is continuous at a point x = c if the following three conditions are met
f(c) is defined
For this link, determine which of the conditions is violated in the examples of discontinuity
x = c

9. "Removing" the Discontinuity
A discontinuity at c is called removable if …
If the function can be made continuous by
defining the function at x = c
or … redefining the function at x = c
Go back to this link, determine which (if any) of the discontinuities can be removed

10. Which of These is Dis/Continuous?
When x = 1 … why or not
if x ≠ 1 and h(x) = 4 if x = 1
if x ≠ 1 and g(x) = 6 if x = 1
if x ≠ 1 and F(x) = 4 if x = 1
Are any removable?

11. Continuity Theorem
A function will be continuous at any number x = c for which f(c) is defined, when …
f(x) is a polynomial
f(x) is a power function
f(x) is a rational function
f(x) is a trigonometric function
f(x) is an inverse trigonometric function

12. Properties of Continuous Functions
If f and g are functions, continuous at x = c Then …
is continuous (where s is a constant)
f(x) + g(x) is continuous
is continuous
f(g(x)) is continuous

13. One Sided Continuity
A function is continuous from the right at a point x = a if and only if
A function is continuous from the left at a point x = b if and only if a b

14. Continuity on an Interval
The function f is said to be continuous on an open interval (a, b) if
It is continuous at each number/point of the interval
It is said to be continuous on a closed interval [a, b] if
It is continuous at each number/point of the interval and
It is continuous from the right at a and continuous from the left at b

15. Continuity on an Interval
On what intervals are the following functions continuous?

16. Intermediate Value Theorem
Given function f(x)
Continuous on closed interval [a, b]
And L is a number strictly between f(a) and f(b)
Then … there exists at least one number c on the open interval (a, b) such that f(c) = L
f(b) c L
f(a) b a

17. Locating Roots with Intermediate Value Theorem
Given f(a) and f(b) have opposite sign
One negative, the other positive
Then there must be a root between a and b a
Try exercises 88, 90, and 92 : pg 100 b

18. Assignment Lesson 2.4A Page 98 Exercises 1 – 49 odd Lesson 2.4B