Transactions of the Cambridge Philosophical Society Volume 21

Transactions of the Cambridge Philosophical Society Volume 21

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 edition. Excerpt: an upper semi-continuous function and the chasm function is a lower semi-continuous function, so that in this case the same is true respectively of the upper and lower functions. Further at a point at which fx) is continuous, U(x) = L(x) + Mf(x) Vol. XXI. No. IX. 34 has necessarily the same kind of semi-continuity as L (#), therefore it is lower semi-continuous; but it is upper semi-continuous everywhere, therefore it is continuous. (II) follows at once from the theory of uniform oscillation, and (III) is obviously a special case of our theorem. 7. Theorem 3a. If in Theorem 3 only the upper lower) limit of the a-series is finite so that M is not finite, then the (a, f)-series oscillates uniformly above (below) so that its upper (lower) function is everywhere upper (lower) semi-continuous. As before we get U(P)u0fQ(P). We now put zero for all the a's before an. The peak function will then be obtained by adding to the peak function of this new series the sum of the first n terms of the (a, /)-series. We thus get II (P) a0f0 + a.f +... + anfn_Y + (un-a0-a1-...-an)fn, where the coefficient of fn is the upper bound of the partial summations of the new series of the a's. Thus un is the upper bound of all the summations after the nth. of the original a-series (1), and hence if we make n describe such a sequence that we get the upper limit for series (1), the coefficient of fn in the preceding inequality will have zero as limit. Hence we get whence the required result follows. It should be noticed that it does not follow because the (a, /)-series oscillates everywhere uniformly above (for example) that it oscillates everywhere finitely. It might, as follows from Theorem 2, diverge properly to--oo above or more points. 8. more

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  • Paperback | 130 pages
  • 189 x 246 x 7mm | 245g
  • English
  • Illustrations, black and white
  • 1236863046
  • 9781236863041