Dini derivative

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A concept in the theory of functions of a real variable. The upper right-hand Dini derivative is defined to be the limes superior of the quotient as , where . The lower right-hand , the upper left-hand , and the lower left-hand Dini derivative are defined analogously. If ( ), then has at the point a one-sided right-hand (left-hand) Dini derivative. The ordinary derivative exists if all four Dini derivatives coincide. Dini derivatives were introduced by U. Dini . As N.N. Luzin showed, if all four Dini derivatives are finite on a set, then the function has an ordinary derivative almost-everywhere on that set.
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