the physicists had to discover it for themselves. The result t h e y n e e d e d is that if lim infn__~(2n+l)(t)[-11" / C > 0 for all t in an interval J, then f is the restriction of an entire function. This condition is equivalent to ~(2n+l)(t)[1/(2n+l) ~ L for all t ( J. A t h e o r e m of Hadam a r d ' s t h e n s h o w s t h a t f(2")(t) satisfies the corres p o n d i n g inequality (with a larger L). Rather than deducing this from H a d a m a r d ' s theorem, I shall simply p r o v e it in the r e q u i r e d form. By Taylor's t h e o r e m with r e m a i n d e r of order 2, if t and t + h belong to J, w e have f(2n)(t) = f(2n-1)(t + K) - f(2n-1)(t) K - -f(2"+l)(t + 0h)K, [0[ < 1, so that ~(2")(t)[ ~ 2[K[-1L2"-I + 1/2KL2"+1. Let h be a positive real n u m b e r less than half the length of J. We m a y suppose that L > 2/h. Then for t E J one of t -+ 2/L J, so we m a y take k = --+2/L to obtain ~(2n)(t)[ ~ 2L 2n, ~(2n)(t)I1/(2n) ~ 21/(2n)L < 2L. I am indebted to Professor Johnson for letting me see the m a n u s c r i p t of [2], and also for helpful comments on an earlier draft of the present paper. References 1. R. P. Boas, A theorem on analytic functions of a real variable, Bull. Amer. Math. Soc. 41 (1935), 233-236. 2. A. Boghossian and P. D. Johnson, Jr., Pointwise conditions for analyticity and polynomiality of infinitely differentiable functions, J. Math. Analysis and Appl. 140 (1989), 301-309. 3. M. J. Hoffman and R. Katz, The sequence of derivatives Amer. Math. Monthly 90 (1983), of a C| 557-560. 4. P. Kolar and J. Fischer, On the validity and practical applicability of derivative analyticity relations, J. Math. Phys. 25 (1984), 2538-2544. 5. W. F. Osgood, Lehrbuch der Funktionentheorie, vol. I, 5th. ed., Leipzig and Berlin: Teubner (1928). 6. A. Pringsheim, Zur Theorie der Taylor'schen Reihe und der analytischen Funktionen mit beschr/inktem Existenzbereich, Math. Ann. 42 (1893), 153-184. 7. H. Salzmann and K. Zeller, Singularit/iten unendlich oft differenzierbarer Funktionen, Math. Z. 62 (1955), 354-367. o 8. I. Vrkoc, Holomorphic extension of a function whose odd derivatives are summable, Czechoslovak Math. J. 35(110) (1985), 59-65. 9. Z. Zahorski, Sur l'ensemble des points singuliers d'une fonction d'une variable r6elle admettant les d6riv6es de tousles ordres, Fund. Math. 34 (1947), 183-245; supplement, ibid. 36 (1949), 319-320. Department of Mathematics Northwestern University Evanston, IL 60208 USA T H E M A T H E M A T I C A L I N T E L L I G E N C E R VOL. 11, N O . 4, 1989 37