3. Definition 1.5.1 (p. 110)
If one or more of the above conditions fails to hold
at C the function is said to be discontinuous.
DEFINITION: CONTINUITY OF A FUNCTION
5. Question 8
EXAMPLE
Solution:
( )
( ) ( )2 2 36
3
x xx x
f x
x
+ −− −
= =
− 3x −
( )
2
2 3
x
f x x where x
= +
∴ = − ≠
1. Given the function f defined as ,
draw a sketch of the graph of f, then by observing
where there are breaks in the graph, determine the
values of the independent variable at which the
function is discontinuous and why each is
discontinuous.
( )
2
6
3
x x
f x
x
− −
=
−
6. •
•
•
•
y
x
Test for continuity: at x=3
1.f(3) is not defined; since the first
condition is not satisfied then f is
discontinuous at x=3.
7. Question 8
2. Given the function f defined as
draw a sketch of the graph of f, then by observing
where there are breaks in the graph, determine the
values of the independent variable at which the
function is discontinuous and why each is
discontinuous.
EXAMPLE
( )
if
if
2
6
3
3
2 3
x x
x
f x x
x
− −
≠
= −
=
10. Question 8Question 8
EXAMPLE
3. Given the function f defined as ,
draw a sketch of the graph of f, then by observing
where there are breaks in the graph, determine the
values of the independent variable at which the
function is discontinuous and why each is
discontinuous.
( )
if
if
2
1
0
2 0
x
f x x
x
≠
=
=
11. 2
0
2
0
1 1
lim
0
1 1
lim
0
0
x
x
x
x
x is a VA
+
−
→
+
→
+
= = +∞
= = +∞
∴ =
VAais0x
0xif
x
1
)x(f
:Graph
2
=
≠=
2
2
1
lim 0
1
lim 0
0
x
x
x
x
y is a HA
→+∞
→−∞
=
=
∴ =
HA
0xatousdiscontinu
isfthensatisfiednotis
conditionondsectheSince
existsnotdoeslim.2
defined;2)0(f.1
:continuityforTest
0x
=
+∞=
=
→
13. Figure 1.5.1 (p. 110)
The figure above illustrates the
function not defined at x=c,
which violates the first condition.
The figure above illustrates that the limit
coming from the right and left both exist
but are not equal, thus the two sided limit
does not exist which violates the second
condition. This kind of discontinuity is called
jump discontinuity.
14. Figure 1.51 (p. 110)
The figure above illustrates that the limit
coming from the right and left of c are
both , thus the two sided limit does
not exist which violates the second
condition. This kind of discontinuity is
called infinite discontinuity.
∞+
The figure above illustrates the function
defined at c and that the limit coming from
the right and left of c both exist thus the two
sided limit exist. But
which violates the third condition.
This kind of discontinuity is called
removable discontinuity.
)x(flim)c(f
cx→
≠
