MAT 1234Calculus I

Section 1.8

Continuity

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Continuous at a point Continuous on an open interval How to demonstrate that a function is not

continuous Intermediate Value Theorem

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Polynomials are “nice” functions when computing limits

Let us look into this “nice” property Other “nice” functions

1. Continuous at a Point

Definition

A function f is continuous at a point a if

)()(lim afxfax

Definition

A function f is continuous at a point a if

)()(lim afxfax

That means…

1.

2.

3.

Example 0 (Section 1.5)

5.1)(lim2

xfx

5.1)(lim2

xfx

5.1)(lim2

xfx

x

y

2

1.5

y=f(x)

Example 0 (Section 1.5)

5.1)(lim2

xfx

5.1)(lim2

xfx

5.1)(lim2

xfx

x

y

2

1.5

y=f(x)

5.1)2( f

Example 1

Explain why the fun. is discontinuous at x=1

1 if 2

1 if 1

1)(

x

xxxf

Note that we need to explain this theoretically. The diagram only serves as an aid.

Example 2

Explain why the fun. is discontinuous at x=1

1 if 4

1 if 1)(

2

xx

xxxf

2. Continuous on an Open Interval

Definition

A function f is continuous on an open interval if it is continuous at every number of the interval.

What if …

If the interval is not open, the definition above breaks down at the end points.

What if …

If the interval is not open, the definition above breaks down at the end points.

A different (modified) definition is required.

We will skip that. Read the text if you are interested.

Common Continuous Functions

Polynomials Rational Functions Root Functions Tri. Functions

Continuous at every no. in their domains

Combinations of Continuous Functions

If f and g is continuous at a, then f+g, f-g, fg, f/g*, cf are also continuous at a.

(*g(a)≠0) ( )y f x

( )y g x

( ) ( )y f x g x

Combinations of Continuous Functions

If g is continuous at a, and f is continuous at g(a), then the composite function

is also continuous at a.f g

Combinations of Continuous Functions

If g is continuous at a, and f is continuous at g(a), then the composite function

is also continuous at a.

In practice, find the composite function and determine its continuity.

f g

Example 32Let ( ) 1 and ( ) .

Is ( ) continuous at 0?

g x x f x x

f g x x

Example 4

For what value of c is the function f continuous at ?

2

1 if 2( )

1 if 2

cx xf x

cx x

2x

Example 4

For what value of c is the function f continuous at ?

2

1 if 2( )

1 if 2

cx xf x

cx x

2x

2

2

(2)

lim ( )

lim ( )

is continuous at 2 if

x

x

f

f x

f x

f x

Expectations2

1 if 2( )

1 if 2

cx xf x

cx x

2

2

(2)

lim ( )

lim ( )

is continuous at 2 if

x

x

f

f x

f x

f x

3. Intermediate Value Theorem

Intermediate Value Theorem

Suppose f is continuous on [a,b] with f(a)≠f(b) and N is between f(a) and f(b)

Then there is a no. c in (a,b) such that

f(c)=N

Intermediate Value TheoremSuppose f is continuous on [a,b] with f(a)≠f(b) and N is between f(a) and f(b)

x

y

b

y=f(x)

a

f(a)

f(b)

N

Intermediate Value TheoremApplications

Use to prove other theorems Use to estimate the roots of equations

• Find a such that f(a)=0

Intermediate Value TheoremApplications

x

y

b

y=f(x)

af(a)

f(b)

c

0

Suppose f(a) and f(b) are with different signs

Then there is a no. c in (a,b) such that f(c)=0

Review: We learned…

Continuous at a point Continuous on an open interval Intermediate Value Theorem Note that in the exam, you may be ask to

state the theorems you learned in this class.