TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A convex function on an open interval has finite left and right derivatives everywhere, with $f_{-}^{'} (u) \leq f_{+}^{'} (u) \leq (f (v) - f (u)) / (v - u) \leq f_{-}^{'} (v) \leq f_{+}^{'} (v)$ for $u < v$

Statement

If $f : I \to R$ is convex on an open interval, then $f_{-}^{'} (c)$ and $f_{+}^{'} (c)$ are finite for every $c \in I$ . Moreover, for $u < v$ in $I$ ,

$f_{-}^{'} (u) \leq f_{+}^{'} (u) \leq \frac{f (v) - f (u)}{v - u} \leq f_{-}^{'} (v) \leq f_{+}^{'} (v) .$

Facts & Assumptions

Given: A convex $f : I \to R$ on an open interval and $u < v$ in $I$ .

[L1]

For a convex function and three ordered points, the three secant slopes satisfy $s (x, y) \leq s (x, z) \leq s (y, z)$ (For a convex function and $x < y < z$ , the three secant slopes satisfy $s (x, y) \leq s (x, z) \leq s (y, z)$ ).

[L2]

The left and right derivatives are the one-sided limits of the difference quotient (The left and right derivatives of a real function as one-sided limits of its difference quotient).

[L3]

A nondecreasing function on an interval has every well-posed one-sided limit, given by the corresponding supremum or infimum (One-sided limits of a monotone function always exist: for $f$ nondecreasing on an interval $I$ and $c \in I$ , $\lim_{x \to c^{-}} f (x) = \sup {f (x) : x \in I, x < c}$ whenever $I$ has points below $c$ , $\lim_{x \to c^{+}} f (x) = \inf {f (x) : x \in I, x > c}$ whenever it has points above $c$ , and these satisfy $\lim_{x \to c^{-}} f (x) \leq f (c) \leq \lim_{x \to c^{+}} f (x)$ ).

Proof

technique · direct

1.1L1L2

For fixed $c \in I$ , the functions $x \mapsto s (x, c)$ on $x < c$ and $x \mapsto s (c, x)$ on $x > c$ are nondecreasing by [L1]; choosing points on both sides of $c$ , [L1] bounds each near $c$ between two fixed finite outer secant slopes.

2.1step 1.1L2L3

The monotone one-sided-limit theorem [L3] therefore supplies finite one-sided limits of these two slope functions at $c$ , and [L2] identifies them respectively with $f_{-}^{'} (c)$ and $f_{+}^{'} (c)$ .

3.1step 1.1step 2.1L1∎

Apply [L1] to $x < u < v$ and let $x \to u^{-}$ , then to $u < v < z$ and let $z \to v^{+}$ ; together with $s (u, v) \leq f_{-}^{'} (v)$ and $f_{+}^{'} (u) \leq s (u, v)$ obtained in the same way, this gives the displayed chain.

Depends on

Dependency tree · 2 levels

Sources

R. Gardner, Convex Functions, Notes 6.6

A convex function on an open interval has finite left and right derivatives everywhere, with f−′(u)≤f+′(u)≤(f(v)−f(u))/(v−u)≤f−′(v)≤f+′(v)f−′​(u)≤f+′​(u)≤(f(v)−f(u))/(v−u)≤f−′​(v)≤f+′​(v) for u<vu<v