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.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
Theorem 1. The sum, difference, product and quotient of continuous functions is continuous provided division by zero is excluded.
Intermediate-value theorem. If a function f is continuous on the interval [a, b], then it takes on every value between f(a) and f(b). In particular it takes on it 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, ... .
3. {x, 2x2, 3x3, ... , nxn}
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
has a limit of 0. The n-th term approaches 0 and meets the requirement of the definition.
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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
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.
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Limit superior. For a sequence of real numbers, the largest accumulation point is called
the limit superior and denoted by lim sup or
. A number
is called the limit superior if
infinitely many terms of the sequence are greater than
- ε for any positive ε, while only a
finite number of terms are greater than
+ 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
. A number
is called the limit inferior if
infinitely many terms of the sequence are less than
+ ε for any positive ε, while only a finite
number of terms are less than
- 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.
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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.
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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:
Theorem 7. A convergent sequence is bounded.
Theorem 8. Every bounded sequence always has a finite lim sup (or
) and lim inf (or
)
and the sequence converges if the two are equal.
Def. Nested intervals. A system of intervals [an, bn] is called nested if
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.
Theorem 9. The closed interval [0, 1] is non-denumerable.
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:
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
since the sequence f1, f2, ..... does not then converge to a continuous function if fn(x) = 0 for
0
x
1/2 and
for 1/2
x
1.
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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:
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)
Mathematics, Its Content, Methods and Meaning.
Natanson. Theory of Functions of a Real Variable