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.
Vol 1A Logic Postscripts
online only include
Proof
by Absurdity alias proof by contradiction
How
the demand for consistency supports the law of the excluded middle
Reality
versus or with the aid of Imagination
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Lipschitz Continuity
Then f(x) is said to be Lipschitz Continuous on this
interval [a,b] if and only if there is a constant K ³ 0
such that
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|f(x2)-f(x1)| £ K·|x2-x1| |
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whenever x1 and x2 are both in the interval [a,b].
Note when
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|f(x2)-f(x1)| £ K·|x2-x1| |
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whenever x1 and x2 are both in the interval [a,b]
then the number K is called a Lipschitz continuity constant for the
function f(x) on the interval [a,b].
Continuity and Equicontinuity
Let e > 0 be given. Suppose K > 0 is a
Lipschitz continuity constant for the function f(x) on the
interval [a,b]. Then |x1-x2| < d = e[1/(K)] implies |f(x1)-f(x2)| £ K·|x1-x2| < e. Thus Lipschitz continuity on an interval implies
continuity at each point x1 on the interval. It furthers
implies equicontinuity on that interval. Recall that a function f(x) is said to be equicontinuous on an interval
[a,b]
if and only if for each e > 0, there exist at
least one d > 0 such that
whenever x1 and x2 are both in the interval [a,b]
and |x1-x2| < d.
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