Continuity. At a point On a open interval On a closed interval Discontinuities

Size: px

Start display at page:

Download "Continuity. At a point On a open interval On a closed interval Discontinuities"

Bertina James
5 years ago
Views:

1 Continuity At a point On a open interval On a closed interval Discontinuities

2 Scenario: From 8 universities, engineering students have been chosen to do 3 things. 1. Build a road to the left (west) of the French Broad River 2. Build a road to the right (east) of the F.B. River 3. Build a bridge across the F. B. River connecting the roads Let s see the results. French Broad River

3 The blue vertical line is the river.

4 First, Furman University, represented by F (x) Will the roads and bridge work? Explain why.

5 First, Furman University, represented by F (x) No!

6 Next, Vanderbilt, represented by V (x) Bridge built of old Martin guitars, but will the roads and bridge work? Explain why.

7 Next, Vanderbilt, represented by V (x) No!

8 Next, University of Georgia, represented by UGA (x) Nice scenic road, but will the roads and bridge work? Explain why.

9 Next, University of Georgia, represented by UGA (x) No!

10 Next, Auburn, represented by A (x) Bridge builders off to football game instead, but will the roads and bridge work? Explain why.

11 Next, Auburn, represented by A (x) No!

12 Next, University of North Carolina, represented by UNC (x) Tar gummed up survey equipment, but will the roads and bridge work? Explain why.

13 Next, University of North Carolina, represented by UNC (x) No!

14 Next, North Carolina State, represented by State (x) Beautiful scenic road to Sandy Mush, but will the roads and bridge work? Explain why.

15 Next, North Carolina State, represented by State (x) No! Now, Explain mathematically

16 Next, Yale University, represented by Y (x) Skull and Bones meeting grabs bridge builders attentions, but will the roads and bridge work? Explain why.

17 Next, Yale University, represented by Y (x) No! lim x a Y (x) Y (a) as Y (a) DNE

18 Next, Whatsamatta U, represented by W (x) Most famous alumni Bullwinkle on hand, but will the roads and bridge work? Explain why.

19 Next, Whatsamatta U, represented by W (x) Yes!!!! because lim x a W (x) = W (a)

20 Continuity at a Point A function f is continuous at c if the following three conditions are met. Think of the Whatsammata University project at x = a.

21 Continuity on an Open Interval A function f is continuous on the open interval (a,b) if it is continuous at each point in the interval. f (x) = 1 (x 2)(x 4) f(x) is continuous on the following intervals: (,2) (2,4) (4, )

22 Continuity on a Closed Interval A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and. The function is continuous from the right at a and continuous from the left at b. lim x 0 + x = 0 = 0 so, continuous on [0, )

23 If a function isn t continuous at a point it is discontinuous there. There are 2 types of discontinuities. 1. Removable only a hole in the function a) Bridge in wrong place or no bridge 2. Non-removable gap on each side of the river infinite discontinuity if vertical asymptote Jump discontinuity if the two one-sided limits aren t equal Let s examine the 7 bad bridges again and identify the type of discontinuity.

24 First, Furman University, represented by F (x) Type of discontinuity? Non-removable jump

25 Next, University of Georgia, represented by UGA (x) Type of Discontinuity? Non-removable infinite discontinuity

26 Next, Vanderbilt, represented by V (x) Type of Discontinuity? Removable discontinuity

27 Next, Auburn, represented by A (x) Type of Discontinuity? Non-removable jump discontinuity

28 Next, University of North Carolina, represented by UNC (x) Type of discontinuity? Non-removable jump discontinuity

29 Next, North Carolina State, represented by State (x) Type of Discontinuity? Non-removable infinite discontinuity

30 Next, Yale University, represented by Y (x) Type of discontinuity? Removable discontinuity

31 Continuity where a piece-wise function comes together (a common continuity problem) Is h(x) continuous at x = 5? 3x 2, x > 5 h(x) = 7, x = 5 x 2 12, x < 5

32 Continuity where a piece-wise function comes together (a common continuity problem) Is h(x) continuous at x = 5? h(x) = 3x 2, x > 5 7, x = 5 x 2 12, x < 5 Does the limit at 5 equal the function value at 5? Does h(5) exist? Does the limit as x approaches 5 exist for h(x)? No!!!!!... h(5) lim x 5 h(x) = 13 removable discontinuity!

33 Continuity where a piece-wise function comes together (a common continuity problem) Is h(x) continuous at x = 5? 3x 2, x > 5 h(x) = 7, x = 5 x 2 12, x < 5 How can we make this function continuous at x = 5?

34 Continuity where a piece-wise function comes together (a common continuity problem) Is h(x) continuous at x = 5? 3x 2, x > 5 h(x) = 7, x = 5 x 2 12, x < 5 How can we make this function continuous at x = 5? Let h(5) = 13

The function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and

The function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous