This paper is concerned with showing differentiability and measure theoretic properties on demand functions. The main results are roughly as follows. (i) Demand is differentiable and the Slutsky equation holds for almost all prices if demand satisfies a Lipschitz condition in income (except possibly for a closed cone of prices of measure zero) and utility is concave. This includes the homothetic case which is given special attention in Section 4. (ii) For the Slutsky equation to indicate demand behavior in the large, it is sufficient that along any given indifference curve the ratio of changes in price to changes in quantity be bounded from zero (Section 5). (iii) Even with almost everywhere differentiable demand derived from continuously differentiable utility, most change in demand does not necessarily take place where the Slutsky equation is valid (Section 5). (iv) By way of proof of Theorems 1-3, it is shown that the maximand in a Lagrange problem is differentiable under appropriate conditions on the function being maximized (Theorem 6, Section 3, and Appendix). (v) For preferences as in (ii) and for almost all wealths, equilibrium is locally unique (Section 6).