Dini Derivative

A Dini derivative of a real-valued function on an interval at is one of the four one-sided limits

(1)

(2)

(3)

and

(4)

For at , they are individually referred to as the upper right, lower right, upper left, and lower left Dini derivatives of at , respectively, and any or all of the values may be infinite.

Finite-valued continuity at a point of a single Dini derivative of a continuous function implies continuity of the other three Dini derivatives of at , equality of the four Dini derivatives, and (usual) differentiability of the function . In addition, the Denjoy-Saks-Young theorem characterizes, outside a set of measure zero, the possible patterns of Dini derivatives of finite real-valued functions defined on intervals and--as corollaries--the Dini derivatives of all monotonic functions and continuous functions defined on intervals.

Many other important properties of Dini derivatives have been studied and characterized. Banach (Kannan and Krueger 1996) showed that the Dini derivative of a measurable function with respect to Lebesgue measure is measurable with respect to Lebesgue measure. Moreover, one can easily show that convex functions satisfy some very precise "almost differentiability" conditions with respect to Dini derivatives (Kannan and Krueger 1996).

Unlike the usual derivative of a function , the Dini derivative can sometimes have unexpected properties. An example attributed to Ruziewicz (Kannan and Krueger 1996) shows that the difference of two continuous functions and on an interval may not be constant even if their extended derivatives agree on , with infinite values allowed. Equality of finite ordinary derivatives throughout would instead make their difference constant by the mean-value theorem.