the Real Analysis appendices of Why Slopes
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These Real Analysis appendices continue the decimal viewpoint of limits, continuity and convergence in chapter 14. and this further lesson
Proofs of one-sided theorems could be of interest in the study of 2D topology.
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If you like these appendices to Volume 3, you may also like (a) the foreword of Volume 3 and chapter 14 with its decimal view of limits, (b) Volume 2, Three Skills for Algebra (for its 4 skills, not 3, for algebra), (c) this treatment of Exponents & Radicals Exactly, (d) this geometric treatment of complex numbers, (e) the Euclidean Geometry with a geometric proof of the distributive law for complex numbers, (f) Pattern Based Reason - its logic elements and online postscripts on logic. |
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
Would you like to show others how to be algebra power users? Professor WhySlopes shouts his methods for algebra skill development are likely to work. Try them. They are different.
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
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whenever x1
and x2 are both in
the interval [a,b] Suppose
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Further suppose xj
< wn < xj+1 .
Then all sums of the form
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.
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)].
n
å
j = 0
f(wj)·(xj+1-xj)
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.
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