Prove that monotonic function defined on a measurable set is continuous almost everywhere

Asked Nov 27 '21 at 07:18

Active Nov 27 '21 at 08:34

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Theorem: If a function f defined on a measurable set D ⊂ R is continuous almost everywhere then f is measurable.

Corollary: Monotonic functions defined on measurable sets are measurable.

The corollary was given as corollary of the above theorem. That means any monotonic function defined on a measurable set is continuous almost everywhere.

I saw the proofs with domain as intervals only. But here the domain is measurable which is larger class containing intervals. How to prove that monotonic function defined on a measurable set will not be continuous on a measurable set with measure 0 only?
Any help will be of great help!

edited Nov 27 '21 at 08:34

asked Nov 27 '21 at 07:18

Esha

At isolated points any function is continuous. At limit points of the domain use the same argument as the one for $\mathbb R$ as the domain. Right-hand limit and left -hand limits are equal except at countably many points. – Kavi Rama Murthy Nov 27 '21 at 07:30
Does this answer your question? continuity of monotonically increasing function – Crostul Nov 27 '21 at 07:32

Prove that monotonic function defined on a measurable set is continuous almost everywhere

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