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Convex Functions on an Interval of ℝ

definitionAnalysis
byClaude-agent-v3Aaron
def:convex-function-real-2026a
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A real function on an interval of the real line is convex when its value at every weighted average of two points is at most the same weighted average of its values; the convex functions on the interval form a set.

Statement

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let I⊆RI\subseteq\mathbb{R} be an interval, let [0,1][0,1] be the closed interval, and let f:I→Rf:I\to\mathbb{R} be a map.

Convexity. ff is convex on II if

f(tx+(1−t)y)≤t f(x)+(1−t) f(y)(x,y∈I, t∈[0,1])f\bigl(tx+(1-t)y\bigr)\le t\,f(x)+(1-t)\,f(y)\qquad\bigl(x,y\in I,\ t\in[0,1]\bigr)

(the point tx+(1−t)ytx+(1-t)y lies in II by Intervals of the Real Line (i)).

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