F. Equicontinuity Thm

Equicontinuity

A function f(x) is said to be equicontinuous on an interval [a,b] if and only if for each e > 0, there exist at least one d > 0 such that

|f(x₂)-f(x₁)| < e

whenever x₁ and x₂ are both in the interval [a,b] and |x₁-x₂| < d.

Thereom F.5 [Equicontinuity Theorem] Suppose b > a. Suppose f(x) is continuous at each point in the interval [a,b]. Then for every e > 0 there exists a d > 0 such that for every pair of points x₁ and x₂ in the interval [a,b],

|x₂-x₁| < d implies |f(x₂)-f(x₁)| < e

The proof given below holds in the case of a function of one variable. It can be generalized to provide the demonstration of a theorem on the equicontinuity of functions of several variables.

Proof of Equicontinuity Theorem.

|f(x₁)-f(x₂)|

|f(x₁)-f(x)+[f(x)-f(x₂)]|

|f(x₁)-f(x)|+|f(x)-f(x₂)|

e =

Therefore the set U(x) =

{d Î [0,1] | x₁ & x₂ Î [x-d,x+d]Ç[a,b] Þ |f(x₁)-f(x₂)| £½e}

contains at least one positive number d £ 1. Thus it is a non-empty subset of the interval [0,1]. Recall the arrow Þ means implies. Therefore we can let g(x) = supU(x). Then g(x) is the largest d > 0 such that |f(x₁)-f(x₂)| £ [1/2]e whenever x₁ and x₂ are in the interval [a,b] within a distance d of the point x. Since U(x) contains at least one positive number, we conclude the function value g(x) = d > 0.

Let x₀ be in the interval [a,b]. We wish to show that g(x) is continuous at x₀. Put d₀ = g(x₀) > 0. Suppose the element c of [a,b] satisfies |c-x₀| £ [1/2]d₀. Then c belongs to the interval [x₀-d₀,x₀+d₀]Ç[a,b]. Further the smallest distance of c to the endpoints of this interval is £ d₀-|x₀-c| = d. From the definition of U(x), we can conclude that d > 0 belongs to U(c). Therefore g(c) ³ d = d₀-|x₀-c| = g(x₀)-|x₀-c|.

Now |x-c| £ [1/2] g(x₀) also implies the largest distance of c to the endpoints of the interval [x₀-d₀,x₀+d₀]Ç[a,b] is d₀+|x-c|. Now suppose for the sake of contradiction that g(c) > d₀+|x-c|. Then d = g(c)-|x₀-c| > d₀ belongs to U(x₀) and hence g(x₀) ³ d > d₀ = g(x₀). But the latter is impossible. Therefore g(c) £ d₀+|x-c| = g(x)+|x-c| when |x-c| £ [1/2] g(x₀) and c is in the interval [a,b].

The above inequalities imply that

g(x₀)-|x₀-c| £ g(c) £ g(x₀)+|x₀-c|

when |x-c| £ [1/2] g(x₀). The inequality is equivalent to |g(x₀)-g(c)| £ |x₀-c|. This latter implies that g(x) is continuous at x₀. And thus g(x) is continuous at each point x = x₀ in the interval [a,b]. By definition, g(x) > 0.

From the finite covering theorem, there exists a whole number n > 0 and a set of numbers x₁,x₂,¼,x_n in [a,b] with the property that if x is a number in the interval [a,b] then there exists an whole number j with 1 £ j £ n and |x-x_j| < g(x_j).

Put (eta) h = [1/2]min_{1 £
j £ n}g(x_j). Now suppose x and y are both in the interval [a,b] with |x-y| < h. Then there exists a x_j with |x_j-x| £ g(x_j). This implies |f(x_j)-f(x)| < [1/2]e. It further implies by the triangle inequality that |y-x_j| = |(y-x)+(x-x_j)| £ |y-x|+|x-x_j| < h+g(x_j) £ 2g(x_j) and hence |f(y)-f(x_j)| £ [1/2]e. Finally, we obtain that

|f(x)-f(y)|

=

|f(x)-f(x_j)+(f(x_j)-f(y))|

£

|f(x)-f(x_j)|+|f(x_j)-f(y)|

£

1
2 e+ 1
2 e = e.

Thus |x-y| < h = d implies |f(x)-f(y)| £ e.

The previous proof has a consequence not stated. In the proof, let e = 1. Then there exists x₁,¼,x_n in [a,b] such that if x is in [a,b] then for some j, |x-x_j| £ d and |f(x)-f(x_j)| £ e = 1. The last inequality is equivalent to

-1 £ f(x)-f(x_j) £ 1

The latter implies

-1+f(x_j) £ f(x) £ 1+f(x_j)

Put K = -1+min_{1 £
q £ n} f(x_q) and M = +1+max_{1 £ q
£ n}f(x_q). Then K £ f(x) £ M if x is in the interval [a,b]. This holds since there exist a whole number j ³ 1 such that

K £ -1+f(x_j) £ f(x) £ 1+f(x_j) £ M

Thus the range W of the function f(x) is contained in the finite interval [K,M]. This implies the first conclusion in the following assertion.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Equicontinuity

the Real Analysis appendices of

Why Slopes
and
More Math
Volume 3

Printed in Canada
ISBN 0-9697564-3-7

Equicontinuity

the Real Analysis appendices of

Why Slopes and More Math Volume 3

Printed in Canada ISBN 0-9697564-3-7

Why Slopes
and
More Math
Volume 3

Printed in Canada
ISBN 0-9697564-3-7