● Title: Exploring Continuity of Functions at a Number
Grade Level: High School or College (11th-12th grade)
Objective:
● Students will understand the concept of continuity of a function at a number.
● Students will learn to identify different types of discontinuities.
● Students will explore graphical representations of continuous and discontinuous
functions.
Materials Needed:
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Whiteboard and markers
Graphing calculators or computers with graphing software
Handouts with exercises
Graph paper
Lesson Plan:
Introduction (10 minutes):
Begin by reviewing the concept of functions and their graphs.
Introduce the concept of continuity of a function at a number, emphasizing the
idea that a function is continuous at a point if the graph has no breaks or jumps
at that point.
Discuss the importance of understanding continuity in calculus and its
applications in various fields.
Definition of Continuity (10 minutes):
● Define continuity of a function at a number, emphasizing the three conditions:
● a. The function is defined at the number.
● b. The limit of the function as it approaches the number exists.
c. The limit of the function as it approaches the number equals the value of the
function at that number.
Discuss each condition and illustrate them using graphical representations.
Types of Discontinuities (15 minutes):
● Present examples of different types of discontinuities:
● a. Removable (or point) discontinuity
● b. Jump (or step) discontinuity
c. Infinite discontinuity
Discuss the characteristics of each type of discontinuity and how to identify
them graphically.
Provide examples and guide students through the process of identifying
discontinuities in functions.
Graphical Representation (20 minutes):
Display graphs of various functions on the board or using graphing software.
Select specific functions and guide students through the process of analyzing
their continuity at different points.
Discuss how to visually identify points of continuity and discontinuity on the
graph.
Facilitate exercises where students analyze graphs and determine the continuity
of functions at specific numbers.
Practice Problems (15 minutes):
Distribute handouts with practice problems for students to solve individually or in
pairs.
Include a variety of problems involving functions with different types of
discontinuities.
Encourage students to identify the points of continuity and discontinuity and
classify the types of discontinuities.
Circulate around the classroom to provide assistance and clarification as
needed.
Review the solutions together as a class, discussing any common
misconceptions or difficulties encountered.
Real-World Applications (10 minutes):
Discuss real-world applications of continuity and discontinuity in fields such as
physics, engineering, and economics.
Explore how the concept of continuity is used to model and analyze various
phenomena and make predictions.
Encourage students to consider how understanding continuity can help in
problem-solving and decision-making in these contexts.
Conclusion and Reflection (10 minutes):
Summarize the key concepts covered in the lesson: continuity of functions at a
number and types of discontinuities.
Ask students to reflect on their understanding of these concepts and their
importance in calculus and beyond.
Encourage students to ask any remaining questions or seek clarification on any
concepts covered in the lesson.
Conclude by emphasizing the significance of mastering continuity for further
studies in calculus and mathematics.
Assessment:
● Assess students' understanding through participation in class discussions,
completion of practice problems, and their ability to correctly identify points of
continuity and discontinuity in functions.
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