3.2: CONTINUITY

Objectives:• To determine whether a function is continuous• Determine points of discontinuity• Determine types of discontinuity• Apply the Intermediate Value Theorem

CONTINUITY AT A POINT—NO HOLES, JUMPS OR GAPS!!A function is continuous at a point c if:1.) f(c) is defined

2.) exists

3.) = f(c)

A FUNCTION NEED NOT BE CONTINUOUS OVER ALL REALS TO BE A CONTINUOUS FUNCTION

)(lim xfcx

)(lim xfcx

1. Does f(2) exist?

2. Does exist?

3. Does exist?

4. Is f(x) continuous at x = 2?

Do the same for x= 1, 3, and 4.

)(lim2

xfx

)(lim2

xfx

REMOVABLE DISCONTINUITY(HOLE IN GRAPH) Limit exists at c but f(c)≠ the limit Can be fixed. Set f(c) =

This is called a continuous extension

)(lim xfcx

EXAMPLE: FIND THE VALUES OF X WHERE THE

FUNCTION IS DISCONTINUOUS. STATE IF IT IS A REMOVABLE DISCONTINUITY. IF SO, FIX IT.

3

9)(

2

x

xxg

OTHER TYPES OF DISCONTINUITY JUMP: ( RHL ≠

LHL)

INFINITE:

OSCILLATING

)(lim xfcx

CONTINUITY ON A CLOSED INTERVAL

A function is continuous on a closed interval [a,b] if:1. It is continuous on the open interval

(a,b)

2. It is continuous from the right at x=a:

3. It is continuous from the left at x=b:

)()(lim afxfax

)()(lim bfxfbx

EXAMPLE

216)( xxf

IT IS CONTINUOUS ON ITS DOMAIN. But discontinuous on x values not in the domain.

KEY FUNCTIONS AND WHERE THEY ARE CONTINUOUSPolynomial or Absolute Value Functions

Continuous for all values of x

Rational Functions: f(x)/g(x) Infinite discontinuity where g(x)=0 (or removable if common factor)

Root Functions (even roots): Continuous where ax + b > 0

Exponential Functions:y= ax, a>0

Continuous for all x

Logarithmic Functions: Continuous for all x> 0

baxy

1,0,log aaxy a

A CONTINUOUS FUNCTION IS ONE THAT IS CONTINUOUS AT EVERY POINT IN ITS DOMAIN. IT NEED NOT BE CONTINUOUS ON ALL REALS.

Where are the functions discontinuous? If it is removable discontinuity, fix it!!

103

2)(.4

sin)(.3

4)(.2

1.1

2

xx

xxf

ttg

xxf

xy

FOR PIECEWISE FUNCTIONS… Check to make sure each “piece” is

continuous Check the x values where it changes

functions. Remember, the following must be true to be continuous at x:

)()(lim)(lim afxfxfaxax

WHERE ARE THE FUNCTIONS DISCONTINUOUS? FIX REMOVABLE!

2,2

2,2

2,3

)(.4

2,3

2,12

1)(.3

2

2.2

0,1

0,1)(.1

2

xx

x

xx

xf

xx

xxxf

x

xy

xx

xxxh