ABSTRACT
In the study of real functions, particularly with attention to differentiation properties, the notion of the variation of a function plays a key role. This chapter describes a study of the first and second order symmetric derivatives. The material for the first order case is mostly reproduced from Brian S. Thomson. The second order variation has not appeared elsewhere; it is based on ideas developed in Freiling, Rinne and Thomson. A function with zero variation should be constant. It is easy to see that a function with zero symmetric variation need not be precisely constant. The classical Schwarz theorem asserts that a continuous function must be linear if it has everywhere a vanishing second symmetric derivative. One can allow exceptional sets by adding hypotheses.
