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On the Sum of Differentiable Functions

Published online by Cambridge University Press:  24 October 2008

H. T. Croft
H. T. Croft
Affiliation:
Peterhouse, Cambridge
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Extract

Ryll-Nardzewski has proposed the following problem (New Scottish Book, no. 119). If fn(x) are continuous, differentiable* functions in a closed finite interval, do there always exist constants cn (no cn = 0) (depending on the ), such that converges and is also a continuous differentiable function?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 2 , April 1962 , pp. 225 - 228
Copyright
Copyright © Cambridge Philosophical Society 1962

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References

* Throughout this article,‘differentiable’ means:‘with a finite (two-sided) derivative at every point’; this derivative may be unbounded. Thus is diffierentiable at x = 0; yẏ=⅓ is not.Google Scholar

† We shall assume this condition implicitly in all the work that follows.Google Scholar

‡ More generally, would also display all the relevant phenomena.Google Scholar

§ A's are absolute constants. O(1), o(l) refer to K →∞.Google Scholar