Symmetric and quantum symmetric derivatives of Lipschitz functions

The symmetric derivative of a real valued function f at the real number x is defined to be lim h 0 f (x + h)− f (x − h) 2h when that limit exists, and if additionally x = 0, the quantum symmetric derivative is defined to be lim q 1 f (qx) −f (q−1x) qx −q−1x when that limit exists. An increasing function ϕ :R+ →R satisfying lim h 0 ϕ(h)/h1/2 = 0 defines by {f : |f (x + h) − f (x)| Cf ϕ(h)} a class of continuous functions which we call a Lipschitz class of functions smoother than Lip 1/2. The symmetric derivative and the quantum symmetric derivative are equivalent pointwise everywhere for functions that are in any Lipschitz class smoother than Lip 1/2, but not necessarily for functions that are Lipschitz of order 1/2.  2003 Elsevier Inc. All rights reserved.