3.2: CONTINUITY Objectives: • To determine whether a function is continuous • Determine points of discontinuity • Determine types of discontinuity • Apply the Intermediate Value Theorem
Dec 31, 2015
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
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
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