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Active Learning Materials for Critical Thinking
in a First Course in Real
Analysis
Barbara Shipman, Ph.D.
Associate Professor
and Distinguished Teaching Professor
The University of Texas at
Arlington
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Complete List of
Materials
Click here to go back
home.
1
Functions
1.1 Definitions of "Function" and "Injective"
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Learning Goals:
Build a clear concept of the definitions of "function" and "injective" and root out some
persistent misconceptions.
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Goal:
Find and correct mistakes in interpreting the terms "unique", "function", and
"one-to-one."
1.1.1
Concept Check: Interpretations of
"unique"
1.1.2 Concept Check: Good
definitions?
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Goal:
Revisit the square root and sine functions, in view of a correct
understanding of the nature of function.
1.1.3 Guided Discovery: The square root
function
1.1.4 Guided Discovery: Two sine
functions
1.1.5 Study Project: Derivatives of two sine
functions
1.1.6 Capstone Connection: The square root in complex
analysis
1.1.7 Capstone Connection: Multi-valued functions in complex
analysis
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Goal:
Reinforce comprehension of injectivity via non-standard examples, proof strategies, and
careful reading of the definition.
1.1.8 Concept Check: Is the function
injective?
1.1.9 Concept Check: Strategies for proving
injectivity
1.1.10
Concept Check: Checking definitions
again
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1 Functions
1.2 Surjectivity and Connections with
Injectivity
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Learning Goals:
Accurately apply the definition of "surjective" and connect surjectivity and injectivity in
different contexts.
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Goal:
Take a critical look at surjectivity via non-standard examples and a comparison of proof
strategies.
1.2.1
Concept Check: Is the function surjective?
1.2.2 Concept Check: Was everyone a
hit?
1.2.3
Concept Check: Strategies for proving
surjectivity
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Goal:
Recognize the role of domain
and codomain in determining properties of a function and discover obstructions to
injectivity and surjectivity of a composition.
1.2.4 Study Project: Finding a domain and a
codomain
1.2.5 Study
Project: Properties of a composition of
functions
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Goal:
Study examples of injective and surjective functions that preserve the structure of rings and vector
spaces.
1.2.6 Capstone Connection: Homomorphisms and linear
functions on Z
1.2.7 Capstone Connection: Derivative and
integral functions |
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1.3
Functions
1.3 Inverse Functions and Inverse
Images
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Learning Goals:
Delineate connections and distinctions among the concepts of perfect pairing, inverse function, and
inverse image.
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Goal: Motivate and apply the definitions of "perfect pairing"
and "inverse function" and guard against misconceptions in using these
definitions.
1.3.1 Guided Discovery: Perfect pairings and inverse
functions
1.3.2 Concept Check: Is there a
perfect pairing?
1.3.3 Concept Check: Why is there
not an inverse?
1.3.4 Study Project: The arcsine
function
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Goal:
Perceive the correct meaning of notation used for both inverse function and inverse
image and compare properties of the two
interpretations.
1.3.5 Concept Check: Inverse functions and
inverse images
1.3.6 Concept Check: The preimage of an
image
1.3.7 Study Project: Inverse images and
properties of functions
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Goal:
View families of curves and surfaces and translations of kernels of matrices as inverse images of
functions.
1.3.8 Capstone
Connection: Inverse images and kernels of matrices
1.3.9 Capstone Connection: Families of curves and
surfaces |
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2 Cardinality
2.1 Basic Notions of
Counting
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Learning Goals:
Build the conceptual framework of counting and cardinality for finite and infinite
sets.
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Goal:
Establish basic notions of counting finite sets as a preview to the more intricate concepts of
cardinality.
2.1.1 Guided Discovery: Handing
out cards
2.1.2 Guided Discovery: Counting a disjoint union
of finite sets
2.1.3 Study Project: Functions to a proper
subset
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Goal:
Determine a "correct" definition for comparing the sizes of infinite sets and expose a common
error in using the definition.
2.1.4 Concept Check: Relabeling
doors
2.1.5 Guided Discovery: A dilemma in comparing
quantities
2.1.6 Guided Discovery: More circles or more
squares?
2.1.7 Concept Check:
An orange
tiger
2.1.8 Concept Check: Testing the
definitions
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Goal: Apply
basic concepts of counting in situations that challenge and reinforce a correct understanding of the
definitions.
2.1.9 Study Project: Proper subsets and cardinality
2.1.10 Study Project: Is there a bijection from (0,1) to
[0,1]? |
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2 Cardinality
2.2 A Hierarchy of Infinite
Cardinalities
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Learning Goals:
Apply and extend the conceptual framework of cardinality in the
unintuitive setting of infinite
sets.
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Goal:
Reflect on an unexpected discovery and subtleties of denoting countable and uncountable
sets.
2.2.1 Historical Pathway: "I see it, but I don't believe
it!"
2.2.2 Study Project: Decimals of rational and irrational
numbers
2.2.3 Concept Check: Denoting infinite collections of
sets
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Goal:
Define the concept of one cardinality being larger than another and contrast this with
dimension.
2.2.4 Guided Discovery: Defining
larger cardinality
2.2.5 Concept Check: A tiger that is
not white
2.2.6 Concept Check: Too large to be
covered
2.2.7 Capstone Connection: Cardinality and
dimension
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Goal:
Consider two advanced ideas on the ordering of infinite cardinalities whose resolutions invoke
logical paradoxes.
2.2.8 Historical Pathway: Reflections on
the Continuum Hypothesis
2.2.9 Guided Discovery: Power sets and a hierarchy of
infinities
2.2.10 Study Project: Cardinality of the power set of
N
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3 Axioms and Bounds
3.1 Algebraic, Order and Completeness
Axioms
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Learning Goals: Analyze the structure of the real number system as a
complete ordered field.
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Goal:
Unfold the levels of structure in the algebraic axioms of
R.
3.1.1
Supplement: The algebraic axioms of
R
3.1.2
Capstone Connection: Levels of algebraic structure in
R
3.1.3 Historical Pathway: Hamilton's
quaternions
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Goal:
Define division and subtraction, with a look at their
properties.
3.1.4 Guided Discovery: Division, subtraction, and
notation
3.1.5 Study Project: Properties of division and
subtraction
3.1.6 Study Project: Distributive
properties
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Goal:
Use the algebraic axioms to prove fundamental results about inverses and identities.
3.1.7 Study Project: Uniqueness of identities and
inverses
3.1.8 Study Project: Inverses of sums and products
3.1.9 Study Project: The inverse of an
inverse
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3 Axioms and
Bounds
3.2
Bounds on Sets and
Functions
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Learning Goals: Compare different bounds on subsets of R and extend these concepts to bounds on
functions.
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Goal:
Find and compare properties of different bounds on sets of real numbers.
3.2.1 Concept Check: Which implies the
other?
3.2.2 Concept Check: Does the bound exist?
Is it unique?
3.2.3 Concept Check: Is it in the set?
3.2.4 Concept Check: Less than the
supremum
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Goal:
Uncover conceptual and notational subtleties in working with bounds on functions.
3.2.5 Concept Check: Bounds on
functions
3.2.6 Concept Check: Bounded functions: True or
false?
3.2.7 Concept Check: Bounded functions: Good
notation?
3.2.8 Guided Discovery: Extending the order
relation
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Goal:
Compare bounded functions in different settings, using different proof
strategies.
3.2.9 Study Project: Comparing bounded functions, Part
I
3.2.10 Study Project: Comparing bounded functions, Part
II
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4 Limits of Sequences
4.1 Definition of
Convergence
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Learning Goals:
Define convergence, see it from various viewpoints, and apply it in some classic
settings.
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Goal:
Arrive at a definition of convergence by considering a spectrum of examples and extracting their
pertinent properties.
4.1.1 Guided Discovery: Definition and
terminology of sequences
4.1.2 Concept Check: Recollections on sequences
4.1.3 Concept Check: Testing statements on
convergence
4.1.4 Guided Discovery: How close? How
many?
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Goal:
View convergence from different perspectives and discover a surprising consequence of the
definition.
4.1.5 Guided Discovery: Different patterns of
convergence
4.1.6 Guided Discovery: Equivalent expressions of
convergence
4.1.7 Guided Discovery: Finitely many terms and
convergence
4.1.8 Historical Pathway: Examining early statements of
convergence
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Goal:
Acquire experience in choosing epsilon appropriately for the purpose of the
problem.
4.1.9 Guided Discovery: Uniqueness of
limits
4.1.10 Study Project: How many terms
are negative?
4.1.11 Study Project: Proving and
disproving
convergence
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4 Limits of Sequencs
4.2 Properties and
Applications
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Learning Goals:
Establish basic properties of convergent sequences, with a look forward to other applications in
analysis.
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Goal:
Prove that every convergent sequence is bounded.
4.2.1 Concept
Check: Trapping a mouse
4.2.2 Guided Discovery: Convergence and
boundedness
4.2.3 Concept Check: Summarizing the
results
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Goal:
Apply the algebra of limits correctly.
4.2.4 Concept Check: Algebra of convergent
sequences
4.2.5 Concept Check: Algebra with divergent
sequences
4.2.6 Study Project: Algebra with
sequences that converge to zero
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Goal:
Appreciate how sequences arise as a cornerstone in major parts of real analysis.
4.2.7
Capstone Connection: Defining infinite sum
4.2.8 Capstone Connection: Sequences and Riemann
integrals
4.2.9 Capstone Connection: Sequences of
functions |
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4 Limits of Sequencs
4.3 Sequences that Tend to
Infinity
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Learning Goals:
Put the concept of tending to infinity on a solid mathematical footing, leaving behind
incorrect images from previous experience and from the notation.
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Goal:
Construct a definition of "tending to infinity" in much the same way as one might construct a
definition of "mammal."
4.3.1 Concept Check: Penguin and platypus
sequences
4.3.2 Guided Discovery: Testing possible
definitions
4.3.3 Guided Discovery: How large? How
many?
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Goal:
Gain a new perspective on tending to infinity and extend the concept to properly divergent
sequences.
4.3.4 Guided
Discovery: Equivalent definitions of tending to infinity
4.3.5 Concept Check: Interpreting the
notation
4.3.6 Study Project: Properly divergent
sequences
4.3.7 Study Project: Algebra of properly divergent
sequences
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Goal:
Revisit "indeterminate forms" from calculus in the rigorous setting of analysis.
4.3.8 Study Project: Combining infinite and nonzero
limits
4.3.9 Study Project: Combining infinite and vanishing
limits
4.3.10 Study Project: Extending the algebra of
limits |
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5 Major Theorems on
Sequences
5.1 Monotone
Sequences
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Learning Goals:
Explore special features of monotone sequences and apply the Monotone Convergence Theorem in
defining the Euler number.
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5.1.1 Guided
Discovery: The Monotone Convergence Theorem
5.1.2 Concept Check: Boundedness, convergence, and
monotonicity
5.1.3 Study Project: Convergence to the supremum
5.1.4 Guided Discovery: A definition
of the Euler number
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5 Major Theorems on
Sequences
5.2
Subsequences
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Learning Goals:
Look more deeply into the structure of sequences by way of the behavior and
existence of subsequences.
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Goal: Take a careful look at
the definition and examples of subsequences before proving the major theorems.
5.2.1 Guided
Discovery
: Definition of a
subsequence
5.2.2 Concept Check: How
many subsequences?
5.2.3 Study Project: Subsequences of an unbounded
sequence
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Goal:
Prove and apply a useful result on convergence of subsequences.
5.2.4 Guided Discovery: The Convergent Subsequence
Theorem
5.2.5 Guided
Discovery:
Proving divergence
5.2.6 Study Project: Shuffled sequences
5.2.7 Study Project: Deducing limits from proven
results
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Goal:
Discover two classic theorems that establish existence of convergent subsequences.
5.2.8 Guided Discovery: The Monotone Subsequence Theorem
5.2.9 Guided Discovery: The Bolzano-Weierstrass
Theorem |
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5 Major Theorems on
Sequences
5.3 Cauchy
Sequences
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Learning Goals:
Build a picture of convergence without reference to a limit and appreciate
the value of this viewpoint.
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Goal:
By way of a practical analogy, unravel what it means for a sequence to be
Cauchy.
5.3.1 Concept Check: How close are the terms?
5.3.2 Concept Check: Does there exist H?
5.3.3 Guided Discovery: Cauchy and
convergence
5.3.4 Guided Discovery: Stuck on
Interstate 10
5.3.5 Study Project:
Choosing epsilon and M
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Goal:
Recognize the convenience of Cauchy conditions in place of convergence in selected
applications.
5.3.6 Study Project: The harmonic sequence
and series
5.3.7 Capstone Connection: A Cauchy
Criterion for infinite series
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6 Limits of Functions
6.1 The Limit of a Function at a
Point
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Learning Goals:
Explore the ideas and the subtleties in the definition of the limit of a function at a
point.
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Goal:
Motivate and make the transition from the definition of a limit used in calculus to the definition
of a limit used in analysis.
6.1.1 Concept Check: The limit versus the value at a point
6.1.2 Guided Discovery: Domain versus behavior in
definitions
6.1.3 Guided Discovery: Finding the
condition on the domain
6.1.4 Concept Check: A rigorous definition
6.1.5 Guilded Discovery: The limits of
language
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Goal:
Explore and apply the concept of a limit as used in analysis to bring out its meaning in a variety
of perspectives and examples.
6.1.6 Study Project: Finding delta with different
concavities
6.1.7 Concept Check: Limits of a
function with domain Q
6.1.8 Study Project: Finding and testing
examples
6.1.9 Concept Check: Does the question make
sense?
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Goal:
Enjoy excursions into how ideas in the definition of a limit appear in topology of the real line,
in differential calculus, and in history.
6.1.10 Capstone Connection: Cluster points and topology of
R
6.1.11 Capstone Connection: Notation for
derivatives
6.1.12 Historical Pathway: Early writings on the notion of a
limit
6.1.13 Historical Pathway: Before the end of that
time
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6 Limits of Functions
6.2 The Sequential Criterion for a
Limit
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Learning Goals:
View the limit of a function at a point in the context of sequences and compare
this perspective with the epsilon-delta definition.
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Goal:
Recognize the utility of the sequential formulation of a limit.
6.2.1 Guided Discovery: The Sequential Criterion
for a Limit
6.2.2 Guided Discovery: A classic
example
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Goal:
Compare the epsilon-delta and the sequential approaches in establishing the
existence or non-existence of a limit.
6.2.3 Concept Check: Another classic example
6.2.4 Guided Discovery: Checking the
proofs
6.2.5 Study Project: Extending the
ideas
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Goal:
Address potential points of confusion in using and stating the Sequential
Criterion for a Limit.
6.2.6 Concept Check: Placing parentheses
6.2.7 Concept Check: The Moose
Criterion |
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6 Limits of
Functions
6.3 Properties,
Extensions, and an Application
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Learning Goals:
Expand the notion of a limit at a point to other types of limits, and explore their properties and
relationships.
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Goal:
Discover how arithmetic operations on functions affect the existence and value of a limit.
6.3.1 Concept Check: Limits of combined
functions
6.3.2 Study Project: What may we
conclude?
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Goal:
Create and apply definitions of other types of limits with a view toward the overarching structure
common to all definitions of limits.
6.3.3 Guided Discovery: Other types of limits of
functions
6.3.4 Concept
Check: Testing a familiar statement on
limits
6.3.5 Study Project: Satisfying requirements on
limits
6.3.6 Study Project: Limits of a product of
functions
6.3.7 Study Project: Limits of a sum of
functions
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Goal: Consider
limits in relation to the weaker condition of local boundedness.
6.3.8 Guided Discovery: Bounded on a
neighborhood
6.3.9 Concept Check: Testing the
definition
6.3.10 Guided Discovery: A close look at
language
6.3.11 Study Project: Limits and local
boundedness
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7 Continuity
7.1 Continuity
at a Point and on a Domain
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Learning Goals:
Recognize the subtleties and differences in definitions of continuity and gain confidence in using
them correctly.
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Goal:
Observe how definitions of continuity used in analysis may clash with previously-developed
intuition and produce perhaps surprising outcomes.
7.1.1 Guided Discovery: Interesting consequences of
definitions
7.1.2 Guided Discovery: Comparing definitions of
continuity
7.1.3 Study Project: Positive on an
interval?
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Goal: Establish
a sense in which some discontinuities are "better" than others.
7.1.4
Removable and essential discontinuities
7.1.5 Concept Check: Removable or
essential?
7.1.6 Study Project: Discontinuities of rational
functions
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Goal:
Consider continuity from the perspective of sequences and in relation to arithmetic
operations.
7.1.7 Concept Check: Using the Sequential
Criterion for Continuity
7.1.8 Study Project: Is the function identically
zero?
7.1.9 Concept
Check: Arithmetic with continuous functions
7.1.10 Study Project: Does there exist an
example?
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7
Continuity
7.2 Continuity
on a Closed Bounded Interval
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Learning Goals: Explore
the strong implications of continuity on a closed bounded interval.
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Goal:
Discover and prove boundedness results on the range of a continuous function on
[a,b].
7.2.1 Guided Discovery: Continuous functions on
intervals
7.2.2 Guided Discovery: Proof of the Boundedness
Theorem
7.2.3 Guided Discovery: The Maximum-Minimum
Theorem
7.2.4 Guided Discovery: Proof of the Maximum-Minimum
Theorem
7.2.5 Study
Project: Continuous and non-continuous
examples
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Goal:
Construct an accurate image of the Intermediate Value Theorem and recognize how and when to use it
appropriately.
7.2.6 Guided Discovery: Picturing different scenarios in the
IVT
7.2.7
Concept Check: Consequences of the
Intermediate Value Theorem
7.2.8 Guided
Discovery: Deducing the Intermediate Value Theorem
7.2.9 Study Project: Looking at
the range
7.2.10 Study
Project: Antipodal points on the
equator
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7
Continuity
7.3 Uniform
Continuity
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Learning Goals:
Develop definitions, theorems, and properties of uniform continuity and the Lipschitz
condition.
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Goal:
Motivate and state the definition of uniform continuity and recognize the dependence of this
condition on the domain.
7.3.1 Guided Discovery:
Definition of uniform continuity
7.3.2 Concept Check:
Uniform continuity over different
domains
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Goal:
Prove the Uniform Continuity Theorem by way of a sequential condition for non-uniform
continuity.
7.3.3 Guided Discovery: Preview to the Uniform Continuity
Theorem
7.3.4 Guided Discovery: The Non-uniform Continuity
Criterion
7.3.5 Guided Discovery: Proving the Uniform Continuity
Theorem
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Goal:
Forumlate the Lipschitz condition and compare it with uniform
continuity.
7.3.6 Guided
Discovery: Lipschitz functions
7.3.7 Guided Discovery: Lipschitz functions and uniform
continuity |
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| © Copyright 2009
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"I should let you know that I tried the very first section that you gave me (on unique and
one-to-one) and it did indeed create a good discussion in class. So, after that, my class was on a good track and I
kept the style. I think this was a very important influence. I am sticking to this style of keeping very much in
touch with the class and this is good."
Dr. Anton Betten
Department of Mathematics,
Colorado State University
"I always looked forward to the challenges that show up in class
and the crazy presentations of true/false questions . . . especially the part where I search for a 'white tiger!'"
Charles Nguyen
Click here to see a white
tiger.
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