Introduction. A comprehensive study of the distribution of the values of the derivites and approximate derivates of measurable functions has been given by J. C. Burkhill and U. S. Haslam-Jones.' In particular they show: If the function f is finite and measurable on a measurable set e, then almost everywhere on e a finite approximate derivative exists, or AD+ = AD= a, AD+ = AD_ = a0. A simple example shows that this result does not include all the possibilities for arbitrary functions. Let the interval (0, 1) be divided into two sets el and e2 with ae1 = ie2 = 1. Letf= 1 on el, andf=0 on e2. Then, at the points of el, AD+ =AD =O AD+ =oo, AD= a. At the points of e2 AD+= AD= O. AD+ =oo, AD_ =0. In a later paper2 these authors studied non-measurable functions f on a set e for which there exists measurable functions sp equal to f on all of e except at most a null set. Obviously no such function so exists for the function f of the above example. In the present paper we consider for arbitrary functions f the distribution of values of derived numbers and approximate derived numbers over sets.3 If the function f is finite at each point of an arbitrary set e, then the upper right derived number of f over e, D.f+, is