04/19/23 Perkins

AP Calculus AB

Day 5Section 1.4

Continuityf(x) will be continuous at x = c unless one of the following

occurs:a. f(c) does not exist lim ( )

cxf x

b. does not exist lim ( ) ( )

cxcf x f

c.

6

4

2

-2

5

6

4

2

-2

5

6

4

2

-2

5c c c

lim ( ) lim ( )x xc c

f x f x

Removable Discontinuity A graph with a “hole” in it

Non-removable Discontinuity Any other type

Discuss the continuity of each.1

1. ( )f xx

2 1

2. ( )1

xg x

x

3. ( ) sinh x x

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

Not continuous at x = 0 (V.A.)Non-removable

Continuous function

1 1

1

x x

x

1x

Not continuous at x = 1

Hole in graph at (1,2)

Removable

23 for 24. ( )

for 2

x xf x

ax x

Find so that ( ) is a continuous function.a f x

If x < 2, the function is a parabola. (continuous)

If x > 2, the function is a line. (continuous)

To be continuous, the two sides must also meet when x = 2.

2

lim ( )x

f x

2

lim ( )x

f x

2

2lim 3x

x

12

2

limx

ax

2a

D.S.

D.S.

2 12a 6a

Intermediate Value Theorem

4

2

-2

-4

-5 5

k

If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k.

a b

The red graph has 1 c-value.

Blue has 5 c-values.

Orange has 1 c-value.

Translation:

If you connect two dots with a continuous function, you must hit every y-value between them at least once.

c c

Perkins

AP Calculus AB

Day 5Section 1.4

Continuityf(x) will be continuous at x = c unless one of the following

occurs:a. f(c) does not exist lim ( )

cxf x

b. does not exist lim ( ) ( )

cxcf x f

c.

6

4

2

-2

5

6

4

2

-2

5

6

4

2

-2

5

lim ( ) lim ( )x xc c

f x f x

Removable Discontinuity

Non-removable Discontinuity

Discuss the continuity of each.1

1. ( )f xx

2 1

2. ( )1

xg x

x

3. ( ) sinh x x

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

23 for 24. ( )

for 2

x xf x

ax x

Find so that ( ) is a continuous function.a f x

Intermediate Value Theorem

4

2

-2

-4

-5 5

k

If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k.

a b

The red graph has 1 c-value.

Blue has 5 c-values.

Orange has 1 c-value.

Translation:

If you connect two dots with a continuous function, you must hit every y-value between them at least once.

c c

Intermediate Value Theorem

4

2

-2

-4

-5 5

If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k.