LESSON 1.10
Continuity (Part 2).
EXERCISES.
Find the constant
?
, or the
constants
?
and
?
, such that the
function is continuous on the
entire real line.
1.
?(?) = {
?
3
, ? ≤ 2
??
2
, ? > 2
2.
?(?) =
{
2, ? ≤ 1
?? + ?, − 1 < ? < 3
−2, ? ≥ 3
Verify that the Intermediate
Value Theorem applies to the
indicated interval and find the
value of
?
guaranteed by the
theorem.
3.
?(?) = ?
2
+ ? − 1
,
[0,5]
,
?(?) = 11
4.
?(𝑡) =
𝑥
2
+𝑥
𝑥−1
,
[0, 3]
,
?(?) = 4
HOMEWORK.
Discuss the continuity of the
composite function
ℎ(?) =
?(?(?))
.
1.
?(?) = ?
2
,
?(?) = ? − 1
2.
?(?) =
1
√𝑥
,
?(?) = ? − 1
3.
?(?) = sin ?
,
?(?) = ?
2
Find the constant
?
, or the
constants
?
and
?
, such that the
function is continuous on the
entire real line.
4.
?(?) = {
𝑥
2
−𝑎
2
𝑥−𝑎
, ? ≠ ?
8, ? = ?
LESSON OBJECTIVE.
* In this lesson you will be able
to justify conclusions about
continuity at a point using the
properties of continuity
* You will also be able to
determine intervals over which a
function is continuous.
ESSENTIAL QUESTION(S).
* Write the definition of
continuity with your own words.
NOTES.
PROPERTIES OF CONTINUITY.
The following types of functions are continuous at every point in their domain.
1.
Polynomial functions:
?(?) = ?
𝑛
?