A Theorem on Semi - Continuous Functions
نویسنده
چکیده
RECENTLY G. C. Young* and A. Denjoyf have communicated theorems—those in Denjoy's memoir are of an especially comprehensive character—dealing, in particular, with point sets where the four derivatives of a given continuous function are identical. It is the purpose of this note to treat an analogous problem that arises when "derivative" is replaced by "saltus."$ However, instead of confining ourselves tc "saltus," we prove a more general theorem that applies essentially to all semi-continuous functions.! We preface the proof of this theorem with the following LEMMA. Let fi(x) and f2{x) be two real, single-valued functions, defined in the linear continuum, such that everywhere fi(x) ^ ƒ2 0*0, and moreover, for every fixed real number k, the set Sk of points x where fi(x) ^ k and fc(x) < k is countable. Thenfi(x) andf2(x) are identical except at most in a countable set Proof. Let {kn}, n = 1, 2, •• • oo, be a set of k's everywhere dense in the linear continuum. The set|| S = ©(#&„)> which consists of all of the elements of every /S&n, is also countable. We show that fi(x) = f2(x) for every given x not in S. For tet {kij be a monotone decreasing sequence of kn's having f2(x) as limit. Since x is given as not belonging to S, it must be that fi(x) < kin for every n; for from fi(x) ^ kin and f2(x) < kin, we would conclude that x belonged to 8kin and
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