1. Definition: Continuous A continuous process is one that takes place gradually, without interruption or abrupt change

2. Continuous Functions Any function y = f (x) whose graph can be sketched in one continuous motion without lifting the pencil is an example of a continuous function

3. If a function f is not continuous at a point c, we say that f is discontinuous at c or c is a point of discontinuity of f.

4. Most of the techniques of calculus require that functions be continuous. Remember: A function is continuous at a point, a, if the limit is the same as the value of the function f(a). This function has discontinuities at x=1 and x=2. It is continuous at x = 0, x = 3, and x = 4, because the one-sided limits match the value of the function 1234 1 2

5. jump infinite oscillating Essential Discontinuities: Examples of: Removable Discontinuities: (You can fill the hole.)

6. Continuity at a Point

7. Where are each of the following functions discontinuous?

8. What on earth is letter d? Greatest Integer Function

9. Can we remove a discontinuity? Sometimes… has a discontinuity at. We can write an extended function that is continuous at.. Note: There is another discontinuity at that can not be removed…why? Let’s look at the graph

10. Removing a discontinuity: Note: There is another discontinuity at.

11. Example Find the values of x which f is not continuous, which of the discontinuities are removable? Removable discontinuity is at: Where as x – 1 is NOT a removable discontinuity.

12. Example: Remove the discontinuity of:

13. Essentially, a discontinuity cannot be removed where the graph approaches an _________________________________ Hint: it sounds like you are saying a bad word. A discontinuity CAN be removed where you have a ________________ Hint: you can make one of these with a shovel.

14. Continuity at a Point If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f. If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f.

15. Example Continuity at a Point [-5,5] by [-5,10]

16. Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. Note: A continuous function may have a discontinuity in its graph…what determines whether a function is “continuous” is whether it is continuous at every point of its domain- not whether it has any “breaks” in the graph.

17. Continuous Functions [-5,5] by [-5,10]

18. So…. All polynomial functions are continuous All rational functions are continuous where ever it is defined; it is continuous ON ITS DOMAIN.

19. Properties of Continuous Functions

20. Composite of Continuous Functions

21. Intermediate Value Theorem for Continuous Functions The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a continuous function on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.

22. Intermediate Value Theorem for Continuous Functions