Continuity
(Section 2.6)
Continuity
limit matches function value
Continuity checklist
1. Is the function value f(a)
defined?
2. Does the limit as xa exist?
3. Does the limit match the
function value?
A continuous function:
Graph has no holes, breaks, or jumps.
You could draw it without lifting your
pencil.
When the x-values are close enough
to each other, so are the
corresponding function values.
When all of the answers are YES, i.e.
lim f ( x )  f ( a ) ,
x a
we say f is continuous at a.
Continuity
limit matches function value
f ( x)  1  x  x  x
2
3
4
lim  f ( x )  2
x 1
lim  f ( x )  2
x 1
lim f ( x )  2
x 1
f (1)  2
Limit as x1 exists and matches
function value, so function is
continuous at x=1.
Discontinuity
limit does not match function value
 x
f ( x)  
6  x
if x  2
if x  2
lim  f ( x )  2
x 2
lim  f ( x )  4
x 2
lim f ( x )
x 2
f (2)  4
Jump discontinuity
(at x=2)
Does Not Exist
Discontinuity
limit does not match function value
f ( x) 
1
( x  3)
2
lim  f ( x )  
x 3
lim  f ( x )  
x 3
lim f ( x )  
x 3
f ( 3 ) undefined
Infinite discontinuity
(at x=3)
Discontinuity
limit does not match function value
f ( x) 
( x  2 )( x  5 )
( x  5)
lim  f ( x )  3
x 5
lim  f ( x )  3
x 5
lim f ( x )  3
x 5
f (5)
Removable discontinuity
(at x=5)
undefined
The discontinuity can be removed by
defining f(5) to be 3.
Discontinuity
limit does not match function value
f ( x) 
sin x
x
lim  f ( x )  1
x 0
lim  f ( x )  1
x 0
lim f ( x )  1
x 0
f (0)
Removable discontinuity
(at x=0)
undefined
The discontinuity can be removed by
defining f(0) to be 1.
Discontinuity
limit does not match function value
 x 2  4
f (x)  
1

if x  1
if x  1
lim  f ( x )  3
x 1
lim  f ( x )  3
x 1
lim f ( x )  3
x 1
f (1)  1
Removable discontinuity
(at x=1)
Limit does not
match function
value
The discontinuity can be removed by
redefining f(1) to be 3.
Discontinuity
limit does not match function value
f ( x )  sin
1
x
lim  f ( x )
Does Not Exist
lim  f ( x )
Does Not Exist
lim f ( x )
Does Not Exist
x 0
x 0
x 0
f (0)
Oscillating discontinuity
(at x=0)
undefined
There is no way to “repair” the
discontinuity at x=0.
Continuity
limit matches function value
Continuity checklist
1. Is the function value f(a)
defined?
2. Does the limit as xa exist?
3. Does the limit match the
function value?
A continuous function:
Graph has no holes, breaks, or jumps.
You could draw it without lifting your
pencil.
When the x-values are close enough
to each other, so are the
corresponding function values.
If all of the answers are YES, i.e.
lim f ( x )  f ( a ) ,
x a
then f is continuous at a.
Left and Right Continuity
left-hand or right-hand limit matches function value
Review: Circles
Circle with radius 3,
centered at origin:
f ( x) 
9x
lim f ( x ) 
2
x  y 9
2
2
continuous
at x = 1
x 1
f (1) 
Solve for y:
y   9 x
2
Top half of circle:
y
9 x
right-continuous
at x = -3
left-continuous
at x = 3
2
lim
Bottom half of circle:
y   9x
2
x  3

f (x)  0
f (  3)  0
lim  f ( x )  0
x 3
f (3)  0
8 2 2
8 2 2