1. CONTINUITY

2. DEFINITION: CONTINUITY OF A FUNCTION
If one or more of the above conditions fails to hold
at C the function is said to be discontinuous.

4. 1. Given the function f defined as ,
EXAMPLE
1. Given the function f defined as ,
draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous.
Solution:
Question 8

5. Test for continuity: at x=3 f(3) is not defined; since the first
condition is not satisfied then f is
discontinuous at x=3.

6. 2. Given the function f defined as
EXAMPLE
2. Given the function f defined as
draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous.
Question 8

7. Solution: y x

9. 3. Given the function f defined as ,
EXAMPLE
3. Given the function f defined as ,
draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous.
Question 8
Question 8

10. HA

11. Solution: y 1 -1 x 1
Question 8
Question 8

12. The figure above illustrates that the limit
coming from the right and left both exist
but are not equal, thus the two sided limit
does not exist which violates the second
condition. This kind of discontinuity is called
jump discontinuity.
The figure above illustrates the
function not defined at x=c,
which violates the first condition.

13. The figure above illustrates that the limit
coming from the right and left of c are both , thus the two sided limit does not exist which violates the second condition. This kind of discontinuity is called infinite discontinuity.
The figure above illustrates the function
defined at c and that the limit coming from
the right and left of c both exist thus the two
sided limit exist. But
which violates the third condition.
This kind of discontinuity is called
removable discontinuity.

14. 2 y x 4

15. 2 4 y x 3
Removable Discontinuity

16. 2 4 y x