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Some Remarks on Uniform Convergence
Published online by Cambridge University Press: 20 January 2009
Extract
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