Differentiation of Monotone Functions

It is clear that limy↓x f(y) ≤ limy↓x f(y). Analogously we can define limy↑x f(y) and limy↑x f(y) and we also have limy↑x f(y) ≤ limy↑x f(y). One can verify as in the sequential case that e.g. (1) limy↓x f(y) ≤ A if and only if for all > 0 there exists a δ > 0 such that f(y) < A+ for all y such that 0 < y − x < δ. (2) limy↓x f(y) ≤ A if and only if for all > 0 and δ > 0 there exists an y with 0 < y − x < δ such that f(y) < A+ . From these and other similar properties one sees that limy→x f(y) = A if and only if limy↑x f(y) = limy↑x f(y) = limy↓x f(y) = limy↓x f(y) = A. Now let F : (a, b)→ R. Then the Dini derivates of F at x are defined as