Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J subset of R be intervals. The main result says that if a superposition operator H generated by a function of two variables h : I x J -> R, H(phi)(x) := h(x, phi(x)), maps the set BV(I, J) of all bounded variation functions phi : I -> J into the Banach space BV(I, R) and is uniformly continuous with respect to the BV-norm, then h(x, y) = a(x)y + b(x), x is an element of I, y is an element of J, for some a, b is an element of BV(I, R). (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim