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 of its domain.
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
x=1
2x 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 compositions1.
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)