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The Dirichlet function is Borel measurable and nowhere continuous

Example

The Dirichlet function

$D : = 1_{Q} : R \to R$

is Borel measurable and nowhere continuous.

Facts & Assumptions

Given: The Dirichlet function $D = 1_{Q}$ .

[L1]

The set of rationals is countable and hence Borel, so its indicator is measurable. ( $Q$ is countably infinite, An indicator function is measurable exactly when its set is measurable)

[L2]

Both $Q$ and $R ∖ Q$ are dense in $R$ . (Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable)

Verification

technique · direct

1.1

By [L1], the function $D$ is Borel measurable.

2.1

Let $x \in R$ and let $U$ be any neighbourhood of $x$ . By [L2], the [step 1.1, given, L2] set $U$ contains both a rational point and an irrational point, so $D$ takes both values $1$ and $0$ on $U$ . Therefore $D$ cannot be continuous at $x$ . Since $x$ was arbitrary, $D$ is nowhere continuous.

step 1.1givenL2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Measure Theory, Section 3.2 (standard reference, not scraped)