differentiability of convex functions
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From: Michael Press on 16 Jan 2010 23:39 In article <bb6f22a4-71b2-474f-8e96-aa5c925fbebf(a)s3g2000yqs.googlegroups.com>, "Dave L. Renfro" <renfr1dl(a)cmich.edu> wrote: > DEFINITION: Let f be defined on an interval I. We say that f > is convex on I if whenever x1, x2 belong to I, then > the line segment whose endpoints are (x1,f(x1)) and > (x2,f(x2)) lies on or above {(x,f(x)): x in [x1, x2]}. > [...] > 1. If f is convex on an open interval I, then f is continuous at > each point in I. A linear function f: R -> R satisfies f(x+y) = f(x) + f(y). In A_Primer_of_Real_Functions [1960], MMA, Ralph P. Boas mentions a construction of a discontinuous linear function. The citations are H.Hahn and A. Rosenthal, Set_Functions, University of New Mexico Press, Alburquerque, 1948, pp. 100ff. G.H. Hardy, J.E. Littlewood, and G. Pólya, Inequalities, Camridge University Press, 1934, p. 96. G. Hamel, Eine Basis aller Zahlen und die unstetigen, Lösungen der Funktionalgleichung f(x + y) = f(x) + f(y), Math. Ann. 60 (1905), 459-462. Since a linear function is convex, this seems to contradict the assertion you make. The construction of a discontinuous linear function requires a Hamel basis. -- Michael Press
From: Robert Israel on 17 Jan 2010 15:50 Michael Press <rubrum(a)pacbell.net> writes: > In article > <bb6f22a4-71b2-474f-8e96-aa5c925fbebf(a)s3g2000yqs.googlegroups.com>, > "Dave L. Renfro" <renfr1dl(a)cmich.edu> wrote: > > > DEFINITION: Let f be defined on an interval I. We say that f > > is convex on I if whenever x1, x2 belong to I, then > > the line segment whose endpoints are (x1,f(x1)) and > > (x2,f(x2)) lies on or above {(x,f(x)): x in [x1, x2]}. > > > > [...] > > > > 1. If f is convex on an open interval I, then f is continuous at > > each point in I. > > A linear function f: R -> R satisfies f(x+y) = f(x) + f(y). > In A_Primer_of_Real_Functions [1960], MMA, Ralph P. Boas > mentions a construction of a discontinuous linear > function. The citations are > > H.Hahn and A. Rosenthal, Set_Functions, University of > New Mexico Press, Alburquerque, 1948, pp. 100ff. > > G.H. Hardy, J.E. Littlewood, and G. Pólya, > Inequalities, Camridge University Press, 1934, p. 96. > > G. Hamel, Eine Basis aller Zahlen und die unstetigen, > Lösungen der Funktionalgleichung f(x + y) = f(x) + f(y), > Math. Ann. 60 (1905), 459-462. > > Since a linear function is convex, this seems to > contradict the assertion you make. The construction of > a discontinuous linear function requires a Hamel basis. No. This function is additive, not linear (or convex) in the usual definition. It satisfies f(s x + t y) = s f(x) + t f(y) if s and t are rational. The usual definition of convex requires f(t x + (1-t) y) <= t f(x) + (1-t) f(y) for all real t with 0 <= t <= 1, not just rationals. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Gc on 17 Jan 2010 17:39
On 17 tammi, 06:39, Michael Press <rub...(a)pacbell.net> wrote: > In article > <bb6f22a4-71b2-474f-8e96-aa5c925fb...(a)s3g2000yqs.googlegroups.com>, > "Dave L. Renfro" <renfr...(a)cmich.edu> wrote: > > > DEFINITION: Let f be defined on an interval I. We say that f > > is convex on I if whenever x1, x2 belong to I, then > > the line segment whose endpoints are (x1,f(x1)) and > > (x2,f(x2)) lies on or above {(x,f(x)): x in [x1, x2]}. > > [...] > > > 1. If f is convex on an open interval I, then f is continuous at > > each point in I. > > A linear function f: R -> R satisfies f(x+y) = f(x) + f(y). > In A_Primer_of_Real_Functions [1960], MMA, Ralph P. Boas > mentions a construction of a discontinuous linear > function. The citations are A linear function from R to R is always just a multiplication by a constant. Proof: f(x) = f(x*1) = x*f(1). > H.Hahn and A. Rosenthal, Set_Functions, University of > New Mexico Press, Alburquerque, 1948, pp. 100ff. > > G.H. Hardy, J.E. Littlewood, and G. Pólya, > Inequalities, Camridge University Press, 1934, p. 96. > > G. Hamel, Eine Basis aller Zahlen und die unstetigen, > Lösungen der Funktionalgleichung f(x + y) = f(x) + f(y), > Math. Ann. 60 (1905), 459-462. > > Since a linear function is convex, this seems to > contradict the assertion you make. The construction of > a discontinuous linear function requires a Hamel basis. > > -- > Michael Press |