1. Section 2.2
Objective:
To understand the meaning of continuous functions.
Determine whether a function is continuous or discontinuous.
To identify the points where a given function is discontinuous.

2. . 1. 2.

3. Discontinuity 3. Types of discontinuity
Hole in graph – removable discontinuity – the function is not defined at x=c
Jump discontinuity
Infinite discontinuity
Oscillating discontinuity

4. Discontinuity
4. A function is continuous at x = c means that there is no interruption in the graph at c. The graph is unbroken at c; no holes, jumps , or gaps. 5. Polynomials are continuous at every real number. f(x) = anxn + an-1xn-1 + ……..a2x2 + ax + c where n is a nonnegative integer

5. Class Examples
Determine whether a function is continuous or discontinuous.
Identify the pionts where a given function is discontinuous. 1.

6. Section 3.5 Class Examples1.

7. Section 3.5 Class Examples
1. At x = 1 and x = 3 denominator equal zero At x = 1 and x = 3 vertical asymptotes At x = 1 and x = 3 infinite discontiuities , nonremovable

8. Class examples 2. X=2 infinite discontinuity nonremovable
Determine whether a function is continuous or discontinuous.
Identify the points where a given function is discontinuous. 2.
X=2 infinite discontinuity nonremovable

9. Class example # 3 (a) Infinite discontinuity Vertical Asymptote
Where denominator equals zero
Discontinuous at x = -7
Discontinuous at x = - 3
Infinite discontinuity, nonremovable

11. 4(a) 4a
X = -3 infinite discontinuity, nonremovable

12. Class examples # 4 (b) Infinite discontinuity Vertical Asymptote
Where denominator equals zero

13. Class examples # 4(b) . Infinite discontinuity Vertical Asymptote
Where denominator equals zero
X= 2,-2 infinite discontinuities, nonremovable

14. Class example # 4(a) Removable discontinuity Hole
When you can cancel a factor of the denominator with a factor of the numerator

15. Class example # 6 Removable discontinuity Hole
When you can cancel a factor of the denominator with a factor of the numerator
X =2 hole discontinuity, removable

16. Class Examples 5(b) . X=-3 hole, removable
X=5 infinite discontinuity, nonremovable

17. Class example # 6 Identify points of discontinuity
X=-2, hole, removable
X=2 infinite discontinuity, nonremovable

18. Jump Discontinuity 7(a)
Different values from the left and the right sides
Piecewise function

19. Jump Discontinuity 7(b)
Different values from the left and the right sides
Piecewise function

20. Class example # 5
Jump
Different values from the left and the right sides
Piecewise functions

21. Class Examples #2
Domain x>1 Continuous for all points greater than 1

22. Closure
What are the three types of discontinuity. Give an example of each.

23. y = x2 y →  as x→-  y →  as x→ y = - x2 y → -  as x→- 
End Behavior of polynomial functions
y = p(x) = anxn + an-1xn-1 + an-2xn-2 +……..a2x2 + ax + ao n>0
an positive, n: even
an negative, n: even
y = x2
y →  as x→- 
y →  as x→
y = - x2
y → -  as x→- 
y → - as x→
an: positive, n: odd
an negative, n: odd
y= x3
y→ -  as x→- 
y→  as x→ 
y= - x3
y→  as x→- 
y → -  as x→ 