1. Limits – Continuity and Intermediate Value Theorem
A continuous function, is one that has no gaps or holes along a given domain.

2. Limits – Continuity and Intermediate Value Theorem
A continuous function, is one that has no gaps or holes along a given domain.

3. Limits – Continuity and Intermediate Value Theorem
A continuous function, is one that has no gaps or holes along a given domain.
This function has no gaps, or holes, and continues on in both directions without bound.

4. Limits – Continuity and Intermediate Value Theorem
A continuous function, is one that has no gaps or holes along a given domain.
This function has no gaps, or holes, and continues on in both directions without bound.
I could trace this graph between any ( a , b ) interval without lifting the pencil off of the paper…

5. Limits – Continuity and Intermediate Value Theorem
A continuous function, is one that has no gaps or holes along a given domain.
This function is continuous but only within a restricted domain or a closed interval.
- 2 2

6. Limits – Continuity and Intermediate Value Theorem
- if any of these conditions are false, then the function is not continuous at “c”

7. Limits – Continuity and Intermediate Value Theorem
- if any of these conditions are false, then the function is not continuous at “c”

8. Limits – Continuity and Intermediate Value Theorem
- if any of these conditions are false, then the function is not continuous at “c”
Functions that are not continuous usually have domain values that are undefined or the function is a Sigma or Piecewise function.

9. Limits – Continuity and Intermediate Value Theorem
- if any of these conditions are false, then the function is not continuous at “c”
Functions that are not continuous usually have domain values that are undefined or the function is a Sigma or Piecewise function.
The Intermediate Value Theorem merely states that if I have a continuous function on a closed interval [ a , b ], if I choose a value within that given interval , the function will be continuous at that point.

10. Limits – Continuity and Intermediate Value Theorem
Sigma or Piecewise functions have different definitions on specific intervals.

11. Limits – Continuity and Intermediate Value Theorem
Sigma or Piecewise functions have different definitions on specific intervals.
EXAMPLE :

12. Limits – Continuity and Intermediate Value Theorem
Sigma or Piecewise functions have different definitions on specific intervals.
EXAMPLE :
As you can see, ƒ(x) has two definitions…

13. Limits – Continuity and Intermediate Value Theorem
Sigma or Piecewise functions have different definitions on specific intervals.
EXAMPLE :
As you can see, ƒ(x) has two definitions…
Here is what the graph of this function would look like.

14. Limits – Continuity and Intermediate Value Theorem
Another way to describe if a function is continuous :
if the limit as “x” approaches “c” from the right of “c”= the limit as “x” approached “c” from the left of “c” , then the function is continuous at “c”

15. Limits – Continuity and Intermediate Value Theorem
Another way to describe if a function is continuous :
if the limit as “x” approaches “c” from the right of “c”= the limit as “x” approached “c” from the left of “c” , then the function is continuous at “c”
We have explored limits approaching a value from the left/right to check if they are equal using a table so another example is not necessary…