Limits, Error Control and Continuity
Volume 3, Why Slopes and More Math.
Error control for the evaluation of functions $y=f(x)$ provides a simple
context and motivation for continuity and convergence.
Continuity at Point
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
\[ |x-a| < \delta=\frac 12 \cdot \frac
1{10^{m}}\] implies \[ |f(x)-f(a)| <\epsilon=\frac 12
\cdot\frac1{10^n} \mbox{ (?)} \]
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 delta $\delta$ and epsilon $\epsilon$ above are
employed here in accordance with tradition of some (not all) calculus
texts. For simplicity, the error control tolerances $\epsilon$ and
$\delta$ in the first instance here and below, may be restricted to be
numbers of the form \[\frac 12 \cdot 10^{-k}=\frac 12 \frac1{10^k}\] 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.
Defintion. [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 \[ |x-a| < \delta=\frac
12 \cdot \frac 1{10^{m}} \] implies \[|f(x)-f(a)|
<\epsilon=\frac 12 \cdot\frac1{10^n}\cdot \]
Decimal-Free Form
The decimal-free description or definition of continuity at a point $x=a$
is as follows.
Defintion. [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
$\epsilon_1>0$, there exist a $\delta_1 > 0$ such that \[ |x-a|
< \delta_1 \] implies \[|f(x)-f(a)| <\epsilon_1 \]
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
$\epsilon_1 > 0$, there is an $n>0$ such that $\epsilon_1 >
\frac 12 \cdot\frac1{10^n}=\epsilon$. The decimal-based requirement for
continuity now is satisfied for some $\delta=\frac 12
\cdot\frac1{10^m}$. So the decimal-free version holds with
$\delta_1=\delta=\frac 12 \cdot\frac1{10^m}$.
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 $ \epsilon_1=\epsilon=\frac 12
\cdot\frac1{10^m}$. Then choose $\delta_1>0$ so that the
decimal-free requirement is satisfied. The decimal-based version is
then satisfied if $m>0$ is selected so that $\delta_1\ge
\delta=\frac 12 \cdot\frac1{10^m}$.
Limit of a Function
Suppose $f(x)$ is a function of real numbers $x$ and that it is defined
on an interval containing the number $a$.
Defintion. [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 \[ |x-a| <
\delta= \frac 12 \frac 1{10^m} \] implies \[|f(x)-L| <\epsilon=
\frac 12 \frac 1{10^n} \]
In the latter case, a limit $L$ is said to exist and we write
\[L=\lim_{x\to a} f(x)\]
The in-line expression $\lim_{x\to a} f(x) $ and the displayed
expression \[ \lim_{x\to a} f(x)\] 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
$L=\lim_{x\to a} f(x)$ to exist and not equal $f(a)$. See the chapter
Slope Approximation.
The rest of this chapter can be read lightly in the first instance. The
next sections are not needed in the immediately following chapters.
Jumps and Limited Error Control
In some cases unlimited error control is not possible at the point $x=a$.
It fails in the following case:
There is an $\epsilon > 0$ such that for every $\delta >0$, there
is some $x$ satisfying the condition $|x-a| < \delta$ and
$|f(x)-f(a)| > \epsilon$.
This means as the input $x$ to the function $y=f(x)$ becomes a better
approximation to the number $a$, there is no guarantee the difference
$|f(x)-f(a)|$ will be smaller than the error control target $\epsilon$.
This concept is illustrated by functions whose graphs have a few jumps in
them. The height of the largest jump near a point $x=a$ indicates how
small the target tolerance $\epsilon$ or $\frac12\cdot 10^{-n}$ can be in
the discussion of error control.
Again, unlimited error control is possible in the following
circumstances:
For each target tolerance $\epsilon > 0$, there is a tolerance
$\delta >0$ such that the condition $|x-a| <
\delta$ implies $$|f(x)-f(a)| \le \epsilon $$
These circumstances appear when $f(x)$ is continuous at $x=a$.
Computations on machines with finite accuracy precision arithmetic,
restrict the number $n$ of decimals places that can be accurately
computed. Every computing machine which calculates to finitely many
binary or decimal places, suffers from such a limit. Small
discontinuities in calculations appear, except in those case where exact
arithmetic can be done. For example, on a computing machine which
computes to at most $n_0$ decimal places, the existence of a rule of the
form \[ |x-a| < \frac 12 \frac 1{10^m} \] implies \[] |f(x)-f(a)|
< \frac 12 \frac 1{10^n} \]
governing error cannot be guaranteed for $n \ge n_0$ and can be
considered improbable for most functions evaluated numerical by a
computer. An exception is provided by functions whose numerically values
can be represented (or encoded) exactly on a machine.
On a computing machine which computes to at most $n_0$ decimal places,
the error control of a single addition and multiplication are guaranteed
to only $n_0$ binary (or decimal) places. Digits beyond the $n_0$ place
are uncertain. If several such calculations are done, with numbers in one
calculation being used in the next, errors accumulate and accuracy is
lost. The calculations in question may have to be reorganized to improve
accuracy.
Significant Digit Error Control
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 \[ \frac{ |x-a|}{|a|} < \frac 12 \frac 1{10^m}
\] implies \[ \frac{|f(x)-f(a)|}{|f(a)|} < \frac 12 \frac 1{10^n}
\mbox{ (?)} \]
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 \[ |x-a| < \frac 12 \frac
1{10^m}\] implies \[ \frac{|f(x)-f(a)|}{|f(a)|} < \frac 12 \frac
1{10^n} \mbox{ (?)} \]
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.
Limits of Sequences
Cauchy Sequences
In dealing with real numbers, we assume that each finite and infinite
decimal expansion defines a real number. When two numbers differ by
$\frac 12 \cdot 10^{-k}>0$, their decimal expansions are said to agree
to $k$ decimal places. Convergence of a sequence to a limit $L$ can now
be expressed in terms of decimal numbers or significant digits: For
any whole number $k$, there is a whole number $N$, such that all terms in
the sequence after the first $N$ agree with the limit $L$ to $k$ decimal
places.
Convergence here corresponds to the ability in principle, if not in
practice, to patiently compute a decimal or binary expansion to an
unlimited number of places.
Error control in practice requires a rate of convergence estimate to say
how large $N$ must be to obtain $k$ decimal places. We may distinguish
between convergence arguments which says there is always $N$ and
convergence arguments which give $N$ as an easily-computed function of
$k$ -- convergence in principle versus the desired situation in which the
rate of convergence can be described and computed.
A Cauchy sequence $f(n)$ has the following property: For each whole
number $k$, there is a whole number $N$ with the following property: all
terms in the sequence after the first $N-1$ agree with each other to at
least $k$ decimal places. This property allows us to define and compute
in principle an infinite decimal expansion. This expansion is assumed to
define a unique real number: the limit $L$ of the Cauchy sequence.
Limit of a Sequence}
Suppose $g(n)$ is a function of whole numbers $n>0$. Then
$g(1),g(2),g(3),\ldots, $ form an infinite sequence of points. This
sequence is said to converge to a finite limit if and only if there is a
real number $L$ such that for every positive number $\epsilon =\frac 12
\frac 1{10^k} > 0$ there is an $N$ such that \[ n>N \mbox{ implies
} |g(n)-L| <\epsilon= \frac 12 \frac 1{10^k} \] In the latter case, a
limit $L$ is said to exist and we write \[L=\lim_{n\to\infty} g(n)\] The
decimal-free equivalent form of the foregoing definition would relax the
requirement that $\epsilon= \frac 12 \frac 1{10^k}$.
The precise decimal-based definition of a Cauchy sequence $g(n)$ is as
follows.
For every whole number $k>0$, there exist a whole number $N$ such
that
\[\mbox{$n\ge N$ and $m\ge N$ implies} |g(n)- g(m)| \le\epsilon
=\frac12\frac1{10^k}\cdot\] The equivalent decimal-free description or
definition of a Cauchy-Sequence $g(n)$ is given next.
For every positive real number $\epsilon>0$, there exist a whole
number $N$ such that \[\mbox{$n\ge N$ and $m\ge N$ implies } |g(n)-
g(m)| \le \epsilon.\]
Arithmetic with Infinite Decimal Expansions
Each real number can be regarded as the limit of an infinite decimal
expansion. Arithmetic with real numbers now requires a discussion of the
addition and multiplication etc of infinite decimal expansions. The
latter will involve some limit concepts and/or the discussion of the
continuity of arithmetic operations $+,-,\div$ and $\times$. The result
of these operations on a pair $a$ and $b$ of real numbers with infinite
decimal expansions can be defined as limit of the sequence which results
from performing the corresponding operation on the decimal expansions to
$n$ decimal places of each real number $a$ and $b$, for $n=1,2,3,\ldots$
and so on. The technical details are omitted here.
They are to be found in the appendices below or in the first chapter in
the text Calculus by Lipman Bers, Holt, Rinehart and Winston 1969,
SBN 03-065240-5. This text has been mentioned earlier.
The details describe say how an error in the knowledge of two numbers $a$
and $b$ affect the error in say the (decimal) computation of $a+b$,
$a-b$, $a\cdot b$, $\frac1b$ and $\frac a{b} =a \cdot \frac 1b$. The
omitted details, given in one of the appendices, further imply the
algebraic properties of limits.
The omitted details also imply that the addition, subtraction,
multiplication and division operations on functions (formulas) continuous
at a point yield further functions continuous at a point provided
division by zero is avoided. The foregoing implies the continuity of many
functions based on the operations involving simpler continuous functions.
Some Old Real Player Videos
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[Play
Video] 4.5 minutes: Algebraic View of Limits. Example
involving sums and quotients.
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[Play
Video] 5.5 minutes: Limits and Error Control for Linear
Expressions
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[Play
Video] 2.75 minutes: Error Control to N decimal Places, say 5 or
10.
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[Play
Video] 3.25 minutes: Limits as Error Control for an unlimited
number of decimal places.
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