WEEKLY LEARNING ACTIVITY SHEET
Basic Calculus, Grade 11, Quarter 3, Week 3
CONTINUITY OF FUNCTIONS
Most Essential Learning Competencies
1.
2.
3.
4.
illustrate continuity of a function at a number (STEM_BC11LCIIIc-1);
determine whether a function is continuous at a number or not (STEM_BC11LCIIIc-2);
illustrate continuity of a function on an interval (STEM_BC11LCIIIc-3); and
solve problems involving continuity of a function (STEM_BC11LCIIId-3)
Specific Objectives
At the end of the lesson, the learner shall be able to:
1.
2.
3.
4.
illustrate continuity of a function at a point;
determine whether a function is continuous at a point or not;
illustrate continuity of a function on an interval; and
determine whether a function is continuous on an interval or not.
Time Allotment:
4 hours
Key Concepts
CONTINUITY AT A POINT
Definition
A function
a)
b)
c)
is said to be continuous at
exists;
exists; and
if the following three conditions are satisfied:
.
If at least one of these conditions is not met,
is said to be discontinuous at
Consider the graph of the function.
•
The function is continuous at
.
•
The function is discontinuous at
.
•
The function is discontinuous at
.
•
The function is continuous at
.
.
Illustration 1. Consider the function
. Is the function continuous at
Sol uti on.
• 𝑓
•
𝑥
•
𝑥
Since 𝑓
𝑥
𝑥
continuous at 𝑥
Illustration 2. Consider the function
?
, then 𝑓 is
.
. Is the function continuous at
?
Sol uti on.
•
•
𝟑 𝟏 𝟐 𝟒 𝟏
𝟏
𝟎
𝒇 𝟏
𝟏 𝟏
𝟎
Si nce 𝒇 𝟏 does not exist, then 𝒇
i s di sconti nuous at 𝒙 𝟏.
CONTINUITY ON AN INTERVAL
Consider the graph of the function below.
Determine whether the function is continuous on
the following intervals:
a)
b)
c)
Consider the graph of the function below.
Determine whether the function is continuous on
the following intervals:
a)
]
b) [
One-sided Continuity
a) A function
is said to be continuous from the left at
b) A function
is said to be continuous from the right at
if
if
Theorem 3.2.1
a) Polynomial functions are continuous everywhere.
| | is continuous everywhere.
b) The absolute value function
c) Rational functions are continuous on their respective domains.
d) The square root function
.
√ is continuous on [
Theorem 3.2.2. A function is said to be continuous…
a) Everywhere if
is continuous at every real number. In this case, we also say
continuous on .
b) on
if is continuous at every point in
.
c) on [
if is continuous on
and from the right at .
] if is continuous on
d) on
and from the left at .
] if is continuous on [
].
e) on [
and on
f) on
if is continuous at all
.
g) on [
if is continuous on
and from the right at .
h) on
if is continuous at all
] if is continuous on
i) on
and from the left at .
is
Illustration 1. Determine the largest interval over which
is continuous.
Solution.
Observe that
is not defined when
. That is,
. Hence, the domain
of is
. From Theorem 3.2.1, a rational function is continuous on its domain. Thus, is
continuous over
.
Illustration 2. Determine the largest interval over which the function
is
√
continuous.
Solution.
Observe that
is defined only if
. That is,
. Moreover, for all
√
,
and
. Hence is continuous over
.
√
√
√
Also, when
,
the right at
.
Therefore, for all
and
√
,
[
√
√
. Hence
is continuous from
is continuous.
Activity No. 1 (Investigate Me!)
What you need: Paper and Ball pen
What to do: Determine whether the following functions are continuous at the given value of .
1.
2.
at
at
3.
at
√
4.
⁄
{
at
5.
at
Activity No. 2 (Solve Me!)
What you need: Paper and Ball pen
What to do: Solve the following problems involving continuity on functions. Show your complete ,
clear, and neat solution.
1. Determine the largest interval where
is continuous.
2. Determine the largest interval where
3. Determine if
4. Let
is continuous.
is continuous or not at
Find the value of c so that
{
5. Consider
√
is continuous at
Find the value of c so that
{
is continuous at
Reflection
On a separate sheet of paper, write a short reflective essay (one to two paragraphs) detailing your
experiences in completing the activities. You may summarize the things that you have learned,
their applications in our daily lives, and the things that you enjoy or dislike.
RUBRICS
10 – 9 points
8 – 6 points
5 – 3 points
2 – 0 points
The reflection explains the
student’s own thinking and
learning experiences, as well
as implications for future
learning.
The reflection explains the
student’s thinking about
his/her own learning
experiences.
The reflection attempts to
demonstrate thinking about
learning but is vague and/or
unclear about the personal
learning experiences.
The reflection does not address
the student’s thinking and/or
learning.
Writer:
Reviewers:
CHRISTIAN JAY M. BUSA
ELMER R. ANDEBOR
AMALIA B. RINGOR, DevEdD
Special Science Teacher I
Agusan National High School
STEM Group Head
Agusan National High School
Academic Track Head
Agusan National High School
RUTH A. CASTROMAYOR
ISRAEL B. REVECHE, PhD
Principal IV
SHS Assistant Principal
Agusan National High School
Education Program Supervisor
Butuan City Division SHS Coordinator
KEY TO CORRECTION
Activity No. 1 (Investigate Me!)
1. Continuous
2. Continuous
3. Continuous
4. Not continuous
5. Continuous
Activity No. 2 (Investigate me!)
1. Solut ion: Note that 𝑔 𝑥 is undefined at 𝑥
. Thus, if 𝑥 ≠ , 𝑔 𝑥
𝑥
, which is a polynomial
𝑥
2.
3.
function. Therefore, 𝑔 𝑥
is continuous for all 𝑥
𝑥
Solut ion: Since 𝑥
, for any 𝑥
ℎ is defined on . Moreover, for any 𝑐
ℎ 𝑐
√𝑐
𝑥 𝑐ℎ 𝑥
Hence, ℎ is continuous at .
Solut ion: We have to check the three conditions for continuity of a function.
1. If 𝑥
, then 𝑓
2.
3.
4.
𝑥 0
𝑓 𝑥
𝑥
𝑥
𝑥
𝑥 0
𝑥
𝑥
𝑥
𝑥 0
𝑓 𝑥 .
𝑓
𝑥 0
,
𝑥
𝑥 0
Solut ion:
1. The function is defined at 𝑥
Since 𝑦
3.
𝑥
2.
𝑥
𝑓 𝑥
and its value is 𝑓
𝑥
𝑥
𝑥
𝑥
𝑥
𝑥
𝑐
𝑐
.
.
is continuous at 𝑥
, we have:
𝑓 𝑥
𝑐𝑥
𝑐
𝑐𝑥
𝑥
5.
𝑥
In order to make all three of these the same, we need 𝑐
Thus, 𝑐
.
Solut ion:
1. The function is defined at 𝑥
and its value is 𝑓
𝑐
2. Since 𝑦
𝑥 𝑐 is continuous at 𝑥
, we have:
𝑓 𝑥
𝑥 𝑐
𝑐.
𝑥
𝑥
3. Since 𝑦
𝑥 is continuous at 𝑥
, we have:
𝑓 𝑥
𝑥
𝑥
𝑥
In order to make all three of these the same, we need
Thus, 𝑐
.
𝑐
Balmecada, J. P. et. al. (2016). Basic Calculus Teacher's Guide (1st ed.). Philippines: Department
of Education.
Leithold, L. (1976). The Calculus with Analytic Geometry (3rd ed.). New York: Harper & Row.
References