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Continuity at a point.
Let a be a real numbers.
Suppose f(x) is a real-valued function for which
for some number L (finite). Moreover, suppose f(a) is defined and L = f(a).
then f(x) is continuous at x = a.
The following theorem will help you quickly identify continuous functions in
any calculus course that you take.
Theorem: Continuity at a Point.
- If f(x) and g(x) are continuous at x =
a, and c is a real number, then scalar multiples, c·f(x),
the sum f(x)+g(x), the difference f(x)-g(x),
the product f(x)·g(x)
are also continuous at x = a.
- If f(x) and g(x) are continuous at x =
a, and g(a) ¹ 0, then the reciprocal [1/(g(x))]
and the quotient or ratio [(f(x))/(g(x))] are also continuous at x = a.
The assertions in this theorem are consequences of the previous theorem on the
algebraic properties of limits.
In practice, continuity at x = a implies the limit
can be evaluated by the immediate substitution of x = a in the function f(x).
Example:
lim
x->
5 |
3x+4 = 3(5) + 4 =
19 |
Continuity of a function f(x) at a number a
corresponds to the requirement that the
limit L = f(a). That is, f(a) = limx-> a f(x) In the latter case, limit evaluation
by immediate substitution is possible.
It is possible for
- the limit L = limx-> a f(x) to exist
and not equal f(a), and
- the limit L = limx-> a f(x) to exist
and f(a) while f(a) is undefined.
In the latter case f(x) is not continuous at x = a.
Continuity on an Interval.
A function f(x) is continuous on an interval I if and only if for each point
a in I, the function f(x) is continuous at x = a. (Here is understood that
one sided limits are to be used at included endpoints of the interval I.) Theorem:
Continuity on an Interval.
- If f(x) and g(x) are continuous on an
interval, and c is a real number, then scalar multiples, c·f(x),
the sum f(x)+g(x), the difference f(x)-g(x),
the product f(x)·g(x)
are also continuous on the interval
- If f(x) and g(x) are continuous on an interval
I, and g(x) ¹ 0 for all points x in the
interval I then the reciprocal [1/(g(x))]
and the quotient or ratio [(f(x))/(g(x))] are also continuous
on the interval.
The assertions in this theorem are consequences of the previous theorem on the
algebraic properties of limits.
Continuity at Point Revisited
To explain the idea of continuity of a function y = f(x) at
a point x = a, we ask the following error-control question with b
= f(a): to what number m of places should the decimal
expansions of x and a agree, for the decimal expansion of the
number f(x) to agree with that of b = f(a) to
n-decimal places? That is, given a whole number n, is there an m
such that
| |x-a|
< d = |
½ |
· |
1
10m |
implies
|f(x)-f(a)|
< e = |
½ |
· |
1
10m |
(?) |
|
An affirmative answer requires that agreement of x with a
to m decimal places implies the agreement of f(x) with f(a)
to n decimal places. An affirmative answer says unlimited accuracy and
error control is possible at x = a.
The Greek letters d (delta) and e
(epsilon) above are employed here in accordance with tradition of some (not all)
calculus texts. For simplicity, the error control tolerances e
and d in the first instance here and below, may be
restricted to be numbers of the form ½ ·10-k
= ½ [1/(10k)]. The decimal free
discussion of error control and continuity dispenses with this requirement.
We say a function f(x) is continuous at a point x = a
if and only if unlimited error control is possible there. More formally, we
state the following definition.
Theorem 14.1 [Continuity at a Point] If f(x) is a
real-valued function of a real number x in an interval [c,d],
and a is a number in the interval [c,d] then the function f
is said to be continuous at the number x = a if and only if the
following holds. If for every n, there exist an m such that
| |x-a|
< d = |
½ |
· |
1
10m |
implies
|f(x)-f(a)|
< e = |
½ |
· |
1
10m |
· |
|
Decimal-Free Form
The decimal-free description or definition of continuity at a point x = a
is as follows.
[Continuity at Point] If f(x) is a real-valued function of a
real number x in an interval [c,d], and a is a point
in the interval [c,d] then the function f is said to be
continuous at x = a if and only if the following holds: For every e1
> 0, there exist a d1 > 0 such that
| |x-a|
< d1
implies |f(x)-f(a)|
< e1 |
|
It is easily shown that the decimal-free and decimal-based definitions are
equivalent. The proof of equivalence, better left to a second reading of this
work, follows.
Proof of Equivalence.
To show the decimal-based description implies the decimal-free description of
continuity, observe the following. First given e1
> 0, there is an n > 0 such that e1
> ½·[1/(10n)] = e. The
decimal-based requirement for continuity now is satisfied for some d
= ½·[1/(10m)]. So the decimal-free version holds with d1
= d = ½·[1/(10m)].
Conversely, the other way that is, to show the latter decimal-free form
implies the decimal-based description of continuity, observe the following.
Given m > 0, let e1 = e
= ½ ·[1/(10m)]. Then choose d1
> 0 so that the decimal-free requirement is satisfied. The decimal-based
version is then satisfied if m > 0 is selected so that d1
³ d = ½·[1/(10m)].
Recapitulation - Limit of a Function
- [Play
Video] 4½ minutes: Algebraic View of Limits. Example
involving sums and quotients.
- [Play
Video] 5½ minutes: Limits and Error Control for Linear
Expressions
- [Play
Video] 2¾ minutes: Error Control to N decimal Places, say 5 or
10.
- [Play
Video] 3¼ minutes: Limits as Error Control for an
unlimited number of decimal places.
Suppose f(x) is a function of real numbers x and that it
is defined on an interval containing the number a.
[Limit of a Function] A function f(x) converges to a
finite limit at the point x = a if and only if there is a real
number L such that for every integer n, there is an m such
that
| |x-a|
< d = |
½ |
|
1
10m |
implies |f(x)-L|
< e = |
½ |
|
1
10n |
|
|
In the latter case, a limit L is said to exist and we write
The in-line expression limx-> a
f(x) and the displayed expression
should both be read as the limit as x goes to a of f(x).
Here remember to read f(x) as f of x.
Continuity of a function f(x) at a number a corresponds
to the requirement that the limit L = f(a). But it is
possible for the limit L = limx-> a
f(x) to exist and not equal f(a).
Significant Digit Error Control
- [Play
Video] 4½ minutes: Algebraic View of Limits. Example
involving sums and quotients.
- [Play
Video] 5½ minutes: Limits and Error Control for Linear
Expressions
- [Play
Video] 2¾ minutes: Error Control to N decimal Places, say 5 or
10.
- [Play
Video] 3¼ minutes: Limits as Error Control for an
unlimited number of decimal places.
The question of relative error is related to the unrestricted control of the
number of significant digits in computations: For every n is there
an m such that
|
|x-a|
|a| |
< |
½ |
1
10m |
implies
|
|f(x)-f(a)|
|f(a)| |
< |
½ |
1
10n |
(?) |
|
This question can only be answered when division by zero is avoided. In
numerical calculations, circumstances may suggest what is more important (more
precisely what is feasible): absolute error control or relative error control.
Various error control (or continuity) questions can be based on different
measures of closeness for x and f(x), that is, different
measures of closeness on the domain and range of a function f. For
example, the question of relative error on the domain can also be posed as
follows: for every n is there an m such that
|
|x-a| |
< |
½ |
1
10m |
implies
|
|f(x)-f(a)|
|f(a)| |
< |
½ |
1
10n |
(?) |
|
For addition and subtraction, absolute error control (the first type introduced
in this chapter) is more appropriate than relative error or significant digit
control. For multiplication and division, relative error and significant digit
error control is more appropriate. When there is a mixture of addition or
subtraction with multiplication or division, no simple advice can be offered. A
course on numerical methods could discuss this topic further.
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Calculus Guide
Section Entrance
Real Player Videos
My First Steps
About Calculus
1. Regular First Steps
2. Limits [13]
3. Differentiation Rules[28]
4. Applications of Derivatives [5]
5. Definite Integrals - Preview [5]
6. Integration Applications [6]
Advanced Material
Up
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
Up
Reference Material: - Light
reading for calculus.
Vol 2, Three
Skills for Algebra covers many topics in algebra and logic
that students starting calculus should have mastered or will have to
master sooner or later. Also includes arithmetic review problems to
catch common mistakes.
Vol. 3, Why
Slopes & More Maths, gives starter lessons for differential
and integral calculus.
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volumes 2 and 3 before calculus and during it. |
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For Parents & Teachers: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly mathematics booklets for ages 4-14.
-
Math
Education
Essays (opinions,
possibilities, references)
- POMME, a two
level program for future skill development in
schools and colleges worldwide. Address content &
motivation gaps with ends, values & methods for skill
development to say which way to go, how and why. -
Present Day Curriculum:
(A) Secondary
I Mathematics
consolidate fractions and measurement, skills and
sense consolidation,
(B)
Secondary II Mathematics
year of algebra and proportionality
(C) See too the following:
- Arithmetic
& Number Theory Practices (horribly put, but
useful)
- Algebra and
Logic SubProgram
(well put, extremely useful)
For Avid Readers in School & Out -
Online Books
1. Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3.Why
Slopes & More.Math
1995
Tour their forewords.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide.
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Senior
High School &
Calculus Students
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Free Live Lesson
- Operations with Decimals - Comparison, Subtraction and Long Division
- Click here
to attend.
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For Senior
High School Mathematics & Calculus
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students.
5
wordy Logic
Chapters
4 curious Algebra
Chapters
Words before & besides symbols. A Key Algebra
forward & backwards Chapter
First Calculus
Preview (1st intro)
Four Calculus
Chapters
(2nd intro)
Intro to Complex
Numbers (long)
Intro to Mathematical
Induction (romantic & wordy at first)
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Many More Topics
1. Decimal
Arithmetic Reference!
2. Integers
- Intro to Signed No.s
3. Fractions
- fully explained.
4. Fractions
with Units
5. Number
Theory,
6. Solving
Linear Equations
7 Formulas
Use Forward & backwards -
8. Proportionality,
Back- & For-wards.
9. Logic
Chapters:
10. Euclidean-Geometry
11. Slopes
& Equations of Straight Lines. (Take
I. See take II below)
12. Why
Study Slopes.
13. Maps,
Plans, Similarity & Trig,
(Take II included here)
14. Quadratics:
Starter lessons
15. Polynomials:
Starter lessons
16 Why
Factor Polynomials:
17 Functions
- Forwards & Backwards.
18. Exponents,
Radicals & logs.
19. Complex
Numbers before trig (new advance/ starter lesson)
20. DC
Electric
Circuits Etc
21. Real
Analysis
22. The
Olde Complex No, Trig
& Vector Section.
23. More
Calculus Stuff
- written after Volumes 2 and 3.
More For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- POMME, a two
level program for instruction K1-14
- Math & Logic How-TOs
1. Arithmetic
2. Algebra
3. More Algebra
4. Beginner Geometry
5. More Geometry
6. Calculus
7. Show Work or Logic
These may be too dense for students.
Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic.
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic
Chapters
(leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps,
Plans, Similarity & Trig, to
appear here).
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Skill Development Tips
For All
Standards: (A) Take
care to avoid the domino effect of errors & approximations; (B) Do and
record steps in an manner that allows skill mastery to be seen or
corrected. Anything represent substandard work.
Key Numerical Methods
- To multiply signed numbers, prefix the product of their signs to the product
of their lengths or unsigned parts. The product is negative if the no of
negative sign in it is odd.
- To add signed numbers with like signs, prefix the common sign to the sum of the
lengths.
- To add signed numbers with opposite signs, prefix the sign of the longest to
the difference: length of longest minus length of shortest.
- Should we study roots and powers of real numbers with formulas involving exponential and log.
- How does adding and multiplying points in the plane and rotating the midpoint
of a line segment lead to mastery of complex numbers and the thought-based
development of their properties, all before trig?
- New Axioms for High School Mathematics: In accounting, totals of assets
and debts may be calculated by dividing the assets and debts into
non-overlapping (disjoint) groups and then adding subtotals. In general, sums
(and products) of counts and numbers, positive and negative numbers
included, can be obtained by adding subtotals (and multiplying
subproducts, respectively). These practices may be cast as axioms in
secondary mathematics. Then operations on polynomials are easily implied
justified by these "axioms" and the geometric introduction of column
methods for expanding a products of two sums. While set theory in pure
mathematics may imply the above axioms in university mathematics programs
instruction, an earlier and more accessible explanation based on easily accepted
and understood geometric and counting practices derivation of the above
axioms is possible at the high school for students heading for college programs
in science.
In Volume 2: Prep for Calculus
- What is the difference between saying A if B and saying A if and
only if B. Being aware of the difference will sharpen ye wits.
- What is a chain of reason?
-Are your arithmetic skills OK?
-Have words been missing in the introduction of algebra?
- Can ye talk about numbers & quantities varying apart from or before the
use of letters & functions?
- Do ye know about the forward & backward use of formulas?
-Contrapositive: is that backward use of A if B?
-What is a variable x? Answer before speaking of function f(x) = x.
-What a twist! There are no rules of algebra for subtraction and division. But
if you replace them by addition of -x and multiplication by 1/x, rules of
algebra (properties of arithmetic) can be used.
In Volume 3: Calculus Slowly?
-Why are slopes studied and polynomials factored in high school?
- Volume 3 suggests how to ease or delay algebra shock in
calculus *& beyond. In Calculus, derivatives and integrals introduced and defined by limits, but calculated
without when possible by using differentiation rules forwards and backwards. The
second site calculus section may help in differential calculus.
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