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
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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