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XV.—On a Test for Continuity.
Published online by Cambridge University Press: 15 September 2014
Extract
§ 1. THE usual method of proving that a function defined as the limit of a sequence of continuous functions is continuous is by proving that the convergence is uniform. This method may fail owing to the presence of points at which the convergence is non-uniform although the limiting function is continuous. In such a case it would be necessary to apply a further test, e.g. that of Arzelà (“uniform convergence by segments”).
In some cases the continuity may be proved directly by means of a totally different principle, without reference to modes of convergence at all. It is, in fact, a necessary and sufficient condition for the continuity of a function that it should be possible to express it at the same time as the limit of a monotone ascending and of a monotone descending sequence of continuous functions.
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- Copyright © Royal Society of Edinburgh 1908
References
page 249 note * Throughout this paper the word “continuous” will be used to mean “bounded and continuous” unless the contrary is stated.
page 249 note † A point of non-uniform convergence where the limiting function is continuous is what I call an “invisible point of non-uniform convergence.”
page 249 note ‡ f1, f2, … . are said to form a monotone ascending sequence if
f1, f2, … being functions of any number of variables. The theorem on which the test depends is that “the limit of an ascending sequence of continuous functions is a lower semicontinuous function, and that the limit of a descending sequence of continuous functions is an upper semi-continuous function. A function which is the limit of both an ascending and a descending sequence is therefore both lower and upper semi-continuous, i.e. it is continuous.” It may be added that there are no invisible points of non-uniform convergence, or divergence, in the case of monotone sequences of continuous functions.—Young, W. H., “On Monotone Sequences of Continuous Functions,” Camb. Phil. Soc. Proc., Lent Term, 1908.Google Scholar
page 251 note * It involves the use of the theorem that “the limit of a descending (ascending) sequence of functions continuous at P is a function which is upper (lower) semi-continuous at P.” This, like the previous theorem quoted on p. 249, of which it is a generalisation, will he found proved in the paper in the Camb. Phil. Soc. Proc. above referred to.
page 257 note * Loc. cit.; for proof see Mess. of Math., 1908.
page 258 note * It may be added that this result can be still further generalised. The general result is that the upper (lower) function of a sequence of lower (upper) semi-continuous functions is upper (lower) semi-continuous except at the points of a set of the first category. Further, this is true with respect to the continuum or any perfect set.