the
Real Analysis appendices of
Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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These Real Analysis appendices
continue the decimal
viewpoint of limits, continuity and convergence in chapter 14. and
this further lesson
A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits, Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II
Proofs of one-sided theorems could be of interest in the
study of 2D topology.
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Lipschitz Continuity and Riemann Sums
Theorem H.3 [Riemann
Sums & Lipschitz Functions] Suppose f(x) is defined on an
interval [a,b] and differentiable when a < x <
b. Further suppose for some K > 0 that
| |f(x2)-f(x1)|
£ K·|x2-x1| |
|
whenever x1 and x2 are both in the interval [a,b]. Suppose
whenever x1
and x2 are both in
the interval [a,b] Suppose
| x0 = a
< x1 < x2 < xj
< xn < xn+1
= b. |
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Further suppose xj
< wn < xj+1 .
Then all sums of the form
|
n
å
j = 0
|
f(wj)·(xj+1-xj) |
|
where 0 < xj+1-xj
< d
differ by at most e = K(b-a)d
Note if e = (1/2)10-k given
first, put d = [e/((b-a)K)].
Proof of Theorem. Let K > 0 be a Lipschitz constant for f(x)
on the interval [a,b]. Let e =
[1/4][1/(10k)][1/(|b-a|)].
Put d = [(e)/((b-a)K)].
Then for every pair of numbers u and v in the interval [a,b],
the inequality |u-v|
£ d implies |f(u)-f(v)|
£ 
. The rest of the proof is exactly the same as the previous one.
Remark. Most piecewise continuous functions met in practice, that
is, in calculus courses, are Lipschitz continuous on each interval [a,b]
which does not include a singularity - a point where division by zero occurs.
But in principle, the exceptions are more frequent. This is analogous to the
situation with real numbers, where, in every-day practice and computation, most
people and computing machines handle fractions and finite decimal or binary
expansions, but where, in principle, there are more irrationals than
rationals among the real numbers. In any event, Lipschitz continuous functions
and criteria which identify them provide further examples of continuous
functions and another link to error control or error bounding in computations.
This author is undecided as to whether or not Lipschitz continuity should be
emphasized in first courses on calculus.
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