A convex function on an open interval is continuous

Builds on Slopes of a convex function are increasing Continuous function

Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be convex. Then $f$ is continuous on $I$ .

Remarks

Fix $y \in I$ and points $x < y < z$ in $I$ . By the slope inequalities, the slopes of chords from $y$ are bounded for points near $y$ . So $|f(w) - f(y)| \le C |w - y|$ near $y$ . On a closed interval this fails. The function on $[0, 1]$ with $f(0) = 1$ and $f(x) = 0$ for $x > 0$ is convex but not continuous at $0$ . A convex function on an open interval has one-sided derivatives at every point. Mathlib states continuity as ConvexOn.continuousOn, for open sets.

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Convex function definition
Limit point of a set definition
Slopes of a convex function are increasing statement
Limit of a function definition
Continuity at a point definition
Continuous function definition
A convex function on an open interval is continuous statement

Tag	`011HV2GAWF`
URI	`m2://011HV2GAWF`
Kind	statement, theorem
Writing	W0 stub
MSC	26A51
Status	This copy of the site has no build data. Run `m2 status 011HV2GAWF`.
Uses	Slopes of a convex function are increasing, Continuous function

A convex function on an open interval is continuous

Statement

Remarks

Reading path

Further reading

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