Elementary Properties of Convex Functions

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In this chapter we discuss some properties of convex functions connected with their boundedness and continuity. We start with the following Lemma 6.1.1. Let D ⊂ ℝ N be a convex and open set, and let f : D→ ℝ be a convex function. Then $$ \frac{{f(x) - f(x - nd)}} {n} \leqslant \frac{{f(x) - f(x - md)}} {m} \leqslant \frac{{f(x + md) - f(x)}} {m} \leqslant \frac{{f(x + nd) - f(x)}} {n} $$ for every x ∈ D, d ∈ ℝ N and m, n ∈ ℕ such that 0 < m < n and x ± nd ∈ D.