continuous functions are measurable

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Example

$f : R \to R$ continuous $⟹$ $f$ is measurable
Notice that for all $α \in R$ , we have

f^{- 1} ((α, \infty]) = f^{- 1} ((α, \infty))

is the preimage of an open set. And because $f$ is continuous, this preimage is also open and therefore measurable.

This follows immediately from the inverse image of measurable functions of all borel sets are measurable.

File	Last Modified
Functional Analysis Lecture 9	2025-07-14
sums and products of measurable functions are measurable	2025-07-14

continuous functions are measurable

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