SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

Continuous functions. Sequences. Accumulation point. Limit superior and inferior. Cauchy sequence. Monotonic sequences. Nested intervals. Cantor’s principle. Metric space. Uniform convergence of sequences of functions. Theorems.



Theorems on continuous functions. Following are some theorems on continuous functions (or mappings):


Theorem 1. The sum, difference, product and quotient of continuous functions is continuous provided division by zero is excluded.


Intermediate-value theorem. Suppose that a function f is continuous on the closed interval a ole.gif x ole1.gif b, and that f(a) ≠ f(b). Then, as x varies from a to b, f(x) takes on every value between f(a) and f(b). In particular it takes on its maximum and minimum values in [a, b].


Theorem 2. If a function f is continuous on a closed point set Q then f is bounded i.e. there exists a real number M such that |f(x)| < M for all x ε Q.

 

Theorem 3. A function is continuous if and only if the inverse image of any open set is also open.



Def. Uniformly continuous function. Let R be the set of real numbers. A function f: R → R is said to be uniformly continuous on a point set Q if given any ε > 0 there exists a number δ such that |f(x) - f(y)| < ε whenever |x - y| < δ where x ε Q, y ε Q.


Theorem 4. If a function f is continuous on a closed bounded point set Q, it is uniformly continuous on Q.




Def. Sequence. A sequence is a set of numbers, quantities or elements arranged in a definite order.


Examples of sequences.


1.         2, 4, 6, 8, 10, ..., 2n, ... .


ole2.gif



3.         {x, 2x2, 3x3, ... , nxn}


ole3.gif


ole4.gif


6.         1, 0, 1, 0, 1, 0, ...


7.         any progression (arithmetic, geometric, harmonic)


A sequence can be either finite or infinite. An infinite sequence is non-terminating, there being another term after each term. A finite sequence has only a definite number of terms. The i-th term of a sequence is often denoted by a(i) or ai. Sequences are denoted by notations such as {a1, a2, a3, .... , an, .... }, {an}, (an). A sequence can be viewed as a particular kind of function, a function whose independent variable n ranges over the set of positive integers.



Limit of a sequence. A sequence of numbers {s1, s2, s3, ... , sn, ... } has the limit s if, for any prescribed accuracy, there is a position in the sequence such that all terms after this position approximate s within this prescribed accuracy i.e for any ε > 0 there exists an N such that |s - sn| < ε for all n greater than N. A series of points {p1, p2, p3, ... } has the limit p if, for each neighborhood U of p there is a number N such that pn is in U if n > N.


Example. The sequence


             ole5.gif


has a limit of 0. The n-th term approaches 0 and meets the requirement of the definition.


                                                                                                James and James. Mathematics Dictionary



Theorem 5. If a sequence has a limit, the limit is unique.



Def. Convergent sequence. A sequence that has a limit is said to be convergent. Otherwise, it is said to be divergent.



Def. Accumulation point (or cluster point or limit point) of a sequence. A point P such that there are an infinite number of terms of the sequence in any neighborhood of P.


Example. The sequence


             ole6.gif


has two accumulation points, the numbers 0 and 1.




Bound to a sequence. An upper bound to a sequence of real numbers is a number which is equal to or greater than every number in the sequence. A lower bound to a sequence of real numbers is a number which is equal to or less than every number in the sequence. If a sequence has both an upper bound and a lower bound, it is said to be a bounded sequence. The smallest upper bound is called the least upper bound. The largest lower bound is called the greatest lower bound.

                                                                                                James and James. Mathematics Dictionary 



Limit superior. For a sequence of real numbers, the largest accumulation point is called the limit superior and denoted by lim sup or ole7.gif . A number ole8.gif is called the limit superior if infinitely many terms of the sequence are greater than ole9.gif - ε for any positive ε, while only a finite number of terms are greater than ole10.gif + ε.

Syn. greatest limit, maximum limit, upper limit



Limit inferior. For a sequence of real numbers, the smallest accumulation point is called the limit inferior and denoted by lim inf or lim. A number l is called the limit inferior if infinitely many terms of the sequence are less than l + ε for any positive ε, while only a finite number of terms are less than l - ε.

Syn. least limit, minimum limit, lower limit



Cauchy’s condition for convergence of a sequence. An infinite sequence converges if, and only if, the numerical difference between every two of its terms is as small as desired, provided both terms are sufficiently far out in the sequence. Tech. The infinite sequence s1, s2, s3, ... , sn, ... converges if, and only if, for every ε > 0 there exists an N such


            | sn+h - sn | < ε


 

that for all n > N and all h > 0.

                                                                                                James and James. Mathematics Dictionary




Def. Cauchy sequence. A Cauchy sequence is a sequence where, given any preassigned positive number ε, however small, there exists a point in the sequence (possibly very far out) beyond which the distance between any two selected elements is less than ε. Tech. A Cauchy sequence is a sequence of points P1, P2, ... such that for any ε > 0 there is a number N for which ρ(Pi, Pj) < ε if i > N and j > N, where ρ(Pi, Pj) is the distance between Pi and Pj. If the points are points of Euclidean space, this is equivalent to the sequence being convergent. If the points are real (or complex) numbers, then ρ(Pi, Pj) is | Pi - Pj | and the sequence is convergent if and only if it is a Cauchy sequence.


Syn. Convergent sequence, fundamental sequence, regular sequence.


                                                                                                James and James. Mathematics Dictionary




Monotonic sequences.


Monotonic increasing sequence. A sequence of real terms a1, a2, ... ,an, ... such that an+1 ≥an for all n i.e. a sequence in which the terms either increase or remain the same.


Monotonic decreasing sequence. A sequence of real terms a1, a2, ... ,an, ... such that an+1 ≤ an for all n i.e. a sequence in which the terms either decrease or remain the same.


Theorem 6. Every monotonic (increasing or decreasing) sequence a1, a2, ... ,an, ... with the property that |an| < M (a constant) converges. That is, every bounded monotonic sequence has a limit.




Theorems on limits of sequences. For sequences consisting of numbers the following hold:


ole11.gif


ole12.gif


ole13.gif


ole14.gif


ole15.gif


ole16.gif


ole17.gif



Theorem 7. A convergent sequence is bounded.


Theorem 8. Every bounded sequence always has a finite lim sup (or ole18.gif ) and lim inf (or lim) and the sequence converges if the two are equal.


Def. Nested intervals. A system of intervals [an, bn] is called nested if


             ole19.gif



Cantor’s principle. For every nested system of intervals [an, bn], n = 1, 2, 3, ... there exists one and only one real number common to all the intervals.


Proof



Theorem 9. The closed interval [0, 1] is non-denumerable.


Proof



Def. Metric space. A metric space is an abstract mathematical system, a generalization of a Euclidean space. It consists of a set arbitrary elements, called points, between which a distance is defined. satisfying a set of axioms (axioms which correspond to the essential properties of a Euclidean space). The distance d(x, y) that is defined between “points” x and y of a metric space must have the following properties:


ole20.gif



Examples of metric spaces


● two dimensional space

● three dimensional space

● n-dimensional space

● one dimensional space (i.e. the real line, the set of real numbers)

● set of all rational numbers

● the space C of continuous functions on the interval [0, 1] with distance defined by the formula


            d(f1, f2) = max | f1(x) - f2(x) |


● any surface in its intrinsic measure

● Hilbert space



Def. Complete metric space. A metric space such that every Cauchy sequence converges to a point of the space.


The space of all real numbers (or of all complex numbers) is complete but the space of all rational numbers is not complete. The space of all continuous functions defined on the interval [0, 1] is not complete if the distance between f and g is defined as is


             ole21.gif


since the sequence f1, f2, ..... does not then converge to a continuous function if fn(x) = 0 for 0 ole22.gif x ole23.gif 1/2 and


             ole24.gif


for 1/2 ole25.gif x ole26.gif 1.

                                                                                     James and James. Mathematics Dictionary 


The space of all continuous functions defined on the interval [0, 1] is complete if the distance between f and g is defined as is

 

            d(f, g) = max { | f(x) - g(x) | : x ε [0, 1] } 



Theorem 10. Every Cauchy sequence of real numbers is convergent.



Uniform convergence of sequences of functions. Let f1, f2, ...... , denoted by {fn}, be a sequence of functions from A to B where A, B ε R, the set of real numbers. We say that {fn} converges uniformly to some function f in A if given some ε > 0 there exists a positive integer n0 such that | fn(x) - f(x) | < ε for all n > n0 and all x ε A.


Theorem 11. If {fn} is a sequence of functions which are continuous in A and uniformly convergent to f in A, then f is continuous in A.



For more information on sequences and series see the following:


Sequences and Series


Cantor’s perfect set. We shall now describe the construction of Cantor’s perfect set, a set with a number of remarkable properties.


Start with the closed interval [0, 1] , trisect it at points 1/3 and 2/3 and remove the open middle third (1/3, 2/3), leaving the closed sets [0, 1/3] and [2/3, 1]. Now remove the open middle third from each of the remaining two intervals [0, 1/3] and [2/3, 1] to give four closed sets. Again remove the open middle third from each of these four sets to give eight closed sets. Do this again. Repeat this process indefinitely. The point set that remains after repeating the process indefinitely is called Cantor’s perfect set.


Cantor’s perfect set is a closed set that contains no isolated points and is thus perfect. In addition, it can be shown that it has the cardinality of the continuum.



References

  James and James. Mathematics Dictionary

  Spiegel. Real Variables (Schaum)

  Taylor. Advanced Calculus

  Mathematics, Its Content, Methods and Meaning.

  Natanson. Theory of Functions of a Real Variable



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]