|
YOU are better than YOU think. Show yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. . That leaves room for thought and
refinement.. |
-/[]\-
||
_ / \
|||||||||||||||||||||||||||| .
|
| |
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.
| |
www.whyslopes.com
More Calculus
[ Back ] [ Up ] [ Next ]
For help in calculus, explore
Volumes
2. Three Skills
for Algebra
and 3. Why
Slopes & More Math, and Calculus
Introduction site area. See how to learn or teach key skills and
concepts, some not all.
Calculus Videos
0. First Calculus Preview
0. Triangle Inequality
0 Inequalities
0. Solving Inequalities
1. Distance+Midpt Formulas
1. Function Domains
1.Polynomial Domain+Range
1. Fn: Linear Combinations
1. Fn Composition II
1. Fn Composition I
1. Solving y**n = x**m
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
3. Derivative Motivation
3. Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x) (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3. Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
4. Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5 Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere
To Learn More, visit Volumes 2 and 3.
Advanced Topics
Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity
To Learn More: Visit Real Analysis.
[ Back ] [ Up ] [ Next ]
|