This preview shows pages 1–3. Sign up to view the full content.
Thursday September 22
−
Lecture 6
:
Continuous functions.
(
Refers to Section 2.5
in
your text
)
Students who have mastered the content of this lecture know about
:
The definition of “f is continuous at a”, continuity from the left or right of a, a function being continuous on
an interval, the basic properties of continuous functions.
Students who have practiced the techniques presented in this lecture will be able to
:
determine whether a
simple function is continuous from the definition or its graph, distinguish between removable, infinite and
jump discontinuities
,
apply the basic properties of continuous functions to determine whether a function is
continuous, evaluate the limit of a continuous function or a composition of continuous functions at a given
point.
6.1
Definition – A function
f
(
x
) is
continuous at a point
a
if lim
x
→
a
f
(
x
) =
f
(
a
).
6.1.1
Note – In order that lim
x
→
a
f
(
x
) =
f
(
a
) three things must hold true:
1)
The number
a
must
be in the domain of
f
.
2)
The limit lim
x
→
a
f
(
x
) must exist, i.e. lim
x
→
a
f
(
x
)
must be a number.
3)
The limit lim
x
→
a
f
(
x
) must be
f
(
a
).
6.1.2
Proposition – The function
f
(
x
) is continuous at
a
if and only if
a
belongs to the
domain of
f
and
lim
x
→
a
[
f
(
x
) –
f
(
a
)] = 0.
holds true.
Proof: If and only if means we must prove both “directions”
⇒
and
⇐
.
⇒
Suppose
a
belongs to the domain of
f
and lim
x
→
a
[
f
(
x
) –
f
(
a
)] = 0 holds true.
Then
f
(
a
) =
k
is a number.
⇐
Suppose
a
belongs to the domain of
f
and lim
x
→
a
f
(
x
) =
f
(
a
) is a number.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 6.1.3
Example – The following function
g
(
t
)
is not continuous at
t
= 0 since lim
x
→
0
−
g
(
t
) = 4 while lim
x
→
0
+
g
(
t
) = 1.3 and so does
lim
x
→
0
g
(
t
)
not exist.
This can be seen in the following graph :
6.1.4
Definition – Suppose
f
is defined on an interval containing
a
in the domain of the
f
. If
f
is not continuous at
a
then we say that
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/28/2011 for the course MATH 116 taught by Professor Robertandre during the Spring '11 term at Waterloo.
 Spring '11
 RobertAndre
 Calculus, Continuity

Click to edit the document details