Let f : I → R, where I is an interval of R; if f is convex

Let f : I → R, where I is an interval of R; if f is convex and d belongs to the interior of I, then f is continuous in d. The function f is convex if and only if ∀a, b, c ∈ I, with a < b < c, it is f (b) ≤

c−b b−a f (a) + f (c) . c−a c−a

(1)

Choose δ ∈ R+ such that [d − δ, d + δ] ⊆ I. Let x ∈ I, such that d < x < a + δ. From (1) with a = d − δ, b = d, c = x, we get f (d) ≤

δ x−d f (d − δ) + f (x) , x−d+δ x−d+δ

hence

x−d x−d+δ f (d) − f (d − δ) . δ δ From (1) with a = d, b = x, c = d + δ, we get f (x) ≥

f (x) ≤

x−d d+δ−x f (d) + f (d + δ) . δ δ

(2)

(3)

Since the right-hand side of inequalities (2) and (3) converge to f (d) as x → d, we can conclude that limx→d+ f (x) = f (d). An analogous argument shows that also limx→d− f (x) = f (d).

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