Dini Derivative

Let be a real-valued function defined on an interval and let . The four one-sided limits

(1)

(2)

(3)

and

(4)

are called the Dini derivatives of at . Individually, they are 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.

It turns out that 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 completely characterizes all possible Dini derivatives of finite real-valued functions defined on intervals and--as corollaries--the Dini derivatives of all monotone and continuous functions defined on intervals.

Many other important properties of Dini derivatives have been studied and characterized. Banach showed that the Dini derivative of a Lebesgue measurable function is Lebesgue measurable. 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. One famous example of such is due to Ruziewicz, who showed that the difference of two continuous functions and on an interval may not be constant even if on ; this is due, in part, to the allowance of infinite Dini derivative.