They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions.
The lower Dini derivative, f′−, is defined by where lim inf is the infimum limit.
For only moderately ill-behaved functions, the two extra Dini derivatives aren't needed.
For particularly badly behaved functions, if all four Dini derivatives have the same value (
) then the function f is differentiable in the usual sense at the point t .