Riemann Sums and Lipschitz Continuity
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.
Calculus Appetizers
& Lessons
YOU are better than YOU think. Show yourself how:
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For better work & study skills, read chapters 2 in Three Skills for Algebra. Sooner is better. Good luck. |
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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Caution: Site advice is approximately correct, for some circumstances, not all. . That leaves room for thought and refinement.. |
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