Convex Real Function is Continuous
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Theorem
Let $f$ be a real function which is convex on the open interval $\openint a b$.
Then $f$ is continuous on $\openint a b$.
Proof
From Convex Real Function is Left-Hand and Right-Hand Differentiable, $f$ is left-hand and right-hand differentiable on $\openint a b$.
From Left-Hand and Right-Hand Differentiable Function is Continuous, $f$ is continuous on $\openint a b$.
$\blacksquare$
Also see
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 22$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.16$