1. Sec 2.5: Continuity
Continuous Function
Intuitively, any function whose graph can be sketched over its domain in one continuous motion without lifting the pencil is an example of a continuous function.

2. Sec 2.5: Continuity Continuity at a Point Continuity Test
A function f(x) is continues at a point a if
Example: study the continuity at x = -1

3. Sec 2.5: Continuity Continuity at a Point (interior point)
Continuity Test
A function f(x) is continues at a point a if
Example: study the continuity at x = 4

4. Sec 2.5: Continuity Continuity at a Point (interior point)
Continuity Test
A function f(x) is continues at a point a if
Example: study the continuity at x = 2

5. Sec 2.5: Continuity Continuity at a Point (interior point)
Continuity Test
A function f(x) is continues at a point a if
Example: study the continuity at x = -2

6. Sec 2.5: Continuity Cont a Continuity at a Point Cont from Cont from
A function f(x) is continues at an end point a if
Cont from
left at a
Cont from
right at a

7. Types of Discontinuities.
Sec 2.5: Continuity
Types of Discontinuities.
removable
discontinuity
Which conditions
infinite
discontinuity
Later:
oscillating
discontinuity:
jump
discontinuity

8. Sec 2.5: Continuity

9. Sec 2.5: Continuity

10. Sec 2.5: Continuity

11. Sec 2.5: Continuity
Continuous
on [a, b]

12. Sec 2.5: Continuity Remark
The inverse function of any continuous one-to-one function is also continuous.

13. Sec 2.5: Continuity
Inverse Functions and Continuity
The inverse function of any continuous one-to-one function is also continuous.
This result is suggested from the observation that the graph of the inverse, being the reflection of the graph of ƒ across the line y = x

14. Sec 2.5: Continuity

15. Sec 2.5: Continuity

16. Sec 2.5: Continuity
continuous

17. Sec 2.5: Continuity

18. Sec 2.5: Continuity
Geometrically, IVT says that any horizontal line between ƒ(a) and ƒ(b) will cross the curve at least once over the interval [a, b].

19. Sec 2.5: Continuity The Intermediate Value Theorem N = ƒ(c)
1) ƒ(x) cont on [a,b]
2) N between ƒ(a) and ƒ(b)
c in [a,b]

20. Sec 2.5: Continuity
One use of the Intermediate Value Theorem is in locating roots of equations as in the following example.

21. Sec 2.5: Continuity

22. Sec 2.5: Continuity