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Some Remarks on Uniform Convergence

Published online by Cambridge University Press:  20 January 2009

H. G. Eggleston
H. G. Eggleston
Affiliation:
University College, Swansea.
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A useful test for uniform convergence is that first established by Buchanan and Hildebrandt [4] which is as follows.

(A) “If a sequence fn (x) of monotonic functions converges to a continuous function f(x) in [a, b] then this convergence is uniform.”

In §1 of this paper it is shown that this test is included in a sequence of theorems, each of which establishes a type of uniform convergence. The first is a well-known topological theorem on limit sets, the second is a result on the limits of rectifiable arcs, the third is a generalisation of (A) due to Behrend [3], the fourth is (A) itself, the fifth is a one-sided version of Bendixson's test and the sixth is Bendixson's test.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 10 , Issue 1 , January 1953 , pp. 45 - 52
Copyright
Copyright © Edinburgh Mathematical Society 1953

References

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