2.3 Continuity

1. 2.3 Continuity

2. Definition • A function y = f(x) is continuous at an interior point c if • A function y = f(x) is continuous at a left endpoint a or a right endpoint b if

3. Requirements for continuity at x = c • There are three things required for a function y = f(x) to be continuous at a point x = c. • c must be in the domain of f, ( i.e. f(c) is defined) 2. 3. If any one of the above fails then the function is discontinous at x = c.

4. Example: Find the points of discontinuity

5. Find the points of discontinuity of f(x) = int(x)

6. Types of discontinuity • Removable (also called point or hole discontinuity) • Jump discontinuity • Infinite discontinuity • Oscillating discontinuity

7. Removable • The exists but is not equal to f(c) either because f(c) is undefined or defined elsewhere. I.e. wherever there is a hole. You can “remove” the discontinuity by defining f(c) to be Examples:

8. Jump discontinuity • The function is discontinuous at x = c because the limit as x approaches c does not exist since

9. Infinite discontinuity • Infinite discontinuity occurs at x = c when • I.e. wherever there is a vertical asymptote • Example: f(x) = 1/x is discontinuous at x = 0 since

10. Oscillating discontinuity • The limit of f(x) as x approaches c does not exists because the y values oscillate as x approaches c. • Example: f(x) = sin(1/x) is discontinuous at x = 0 since the y values oscillate between 1 and –1 as x approaches 0

11. Example 1 • Find and classify the points of discontinuity, if any. • If removable, how could f(x) be defined to remove the discontinuity?

12. Example 2 • Find and classify the points of discontinuity, if any. • If removable, how could f(x) be defined to remove the discontinuity?

13. Example 3 • Find and classify the points of discontinuity, if any. • If removable, how could f(x) be defined to remove the discontinuity?

14. Example 4 • Find and classify the points of discontinuity, if any. • If removable, how could f(x) be defined to remove the discontinuity?

15. Example 5 • Find and classify the points of discontinuity, if any. • If removable, how could f(x) be defined to remove the discontinuity?

16. Example 6 • Find and classify the points of discontinuity, if any. • If removable, how could f(x) be defined to remove the discontinuity?

17. Continuous functions • A continuous function is a function that is continuous at every point ofitsdomain. • For example: y = 1/x is a continuous function because it is continuous at every point in its domain. We say that it is continuous on its domain. It is not, however, continuous on the interval [-1,1] for example. • y = |x| is a continuous over all reals (its domain) so it is a continuous function

18. Properties of continuous functions • If f and g are continuous at x = c, then the following are continuous at x = c. • f + g • f – g • fg • f/g provided g(c) is not zero • kf where k is a constant

19. Discuss the continuity of each function • F(x) = x x<1 2 x=1 2x-1 x>1 • G(x) = int (x) • y = x2 + 3

20. Continuity of compositions • If f is continuous at x = c and g is continuous at f(c) then g(f(x)) is continuous at x = c.

21. Discuss the continuity of the compositions 1. • f(g(x)) if f(x) = and g(x) = x2 + 5 3. f(g(x)) if f(x) = and g(x) = x – 1

22. Intermediate Value Theorem • A function y = f(x) that is continuous on [a, b] takes on every y-value between f(a) and f(b) This theorem guarantees that if a function is continuous over an interval then the graph will be connected over the interval – no breaks, jumps or branches (f(x) = 1/x)