Convex Real Function is Continuous

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): $\S 12.16$
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