Calculus I
Chapter 2(6)
Continuity
Limits: Piece Functions
x  2
x

f ( x )  2 x  2  x  1
2
x 1

Lim
= -2
Lim
Lim
x 2
x 2
x 2
Lim
= 2 (middle)
= -4 (Middle)
Lim
= 2 (bottom)
= No Limit
Lim
= 2 (are equal)
(top)
(not equal)
x 1
x 1
x 1
Limits: Piece Functions
x  3
 x
 4
x  3

f ( x)  
 0 2 x 3
 x  3
x3
Lim
=3
Lim
=0
Lim
= No Lim
Lim
=0
Lim
= No Limit
Lim
=0
x  3
x  3
x  3
x 3
x 3
x3
Continuity at a point
Three things can happen at a point on a curve:
The curve is continuous
Solid – no breaks
The curve is missing a
single point
The curve has a “vertical break”
Continuity at a point
Mathematically limits and function
definitions are involved in each case.
The curve is continuous
The defined point
Equals the limit
The curve is missing a single point
The defined point
Does not equal the limit
The curve has a “vertical break”
There is no limit
Requirements for Continuity
The function must be
defined at the point
The limit must exist
The limit = the definition
f (c) exists
Lim f ( x) exists
x c
Lim f ( x)  f (c)
x c
Graphing
Example
At x = 2
At x = 1
Definition:
0
Definition:
Limit
-1
Limit:
Equal:
No
Equal:
Not Continuous
At x = 4
Definition:
0
None
Limit:
0
No
Equal:
0
1
Not Continuous
Continuous
Algebraic Functions and Continuity
All polynomials are continuous
Two Trig functions are continuous
Sine
Cosine
Fractions are continuous where the denominator ≠ 0
Radicals are continuous where the inside > 0
Piece functions must be checked where the “breaks” are.
Types of discontinuities
Removable – only a single point is missing
Non-Removable – there is no limit at the point
(there is a jump between the two sides)
Give the points (if any) where the
function is not continuous.
f ( x)  7 x  tan x
2
This is not continuous
at 2  n
because of Tangent
These are non-removable
f ( x)  3x  6 x
2
This is not continuous
on 0 < x < 2
This is non-removable
3x 2  6 x  0
3x( x  2)  0
x  0 and x  2
No
Yes
0
Yes
2
x  3
 x
 3
x  3

f ( x)  
 0 2  x  3
 x  3
x3
Definition:
Limit
Equal:
At x = 3
At x = 1
At x = -3
3
No Lim
No
Not Continuous
(non-removable)
0
Definition:
Limit:
0
Limit:
Equal:
Yes
Definition:
Continuous
Not
0
Equal:No
Not Continuous
(removable by
adding one point)
Find any discontinuities
3x  6
f ( x)  2
x  x6
x2  x  6  0
( x  2)( x  3)  0
x  2 and x  3
3( x  2)
f ( x) 
( x  2)( x  3)
x = 2 is a removable
discontinuity
x = -3 is a non-removable
discontinuity