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where A± is a number bounded as rn, sn -» oo and fixed when rn and sn are fixed modulo 4. It is easy to prove that if x is irrational, xn -* x in such a way that \x — xn\= 0(s~2), and/is differentiable at x, then limn^xB^ exists with value 0. As shown in the Lemma below, it is possible to find an infinite sequence of rationals [xn] with xn -» x, rnsn = 0 (mod2) and \x — xn\< l/s2. Hence, it is possible to find an infinite sequence of rationals xn all satisfying one of the further restraints, sn = 1 (mod 4), s„ = 3 (mod 4), rn = 1 (mod 4), rn = 3 (mod 4). With the proper choice of sign (say + ) we can then assume, by Lemma 3, that A+ ¥= 0 and fixed. From (i) this leads to the contradiction that | B*1 is bounded away from 0, so/ cannot be differentiable at x.

Correction to: “The differentiability of Riemann’s function” [Proc. Amer. Math. Soc. 34 (1972), 463–468; MR0308337 (46 #7451)]