= Finite modification of the identity on the real line
{title2=$|\{x:f(x)\ne x\}|<\infty$}
A finite modification agrees with the <identity map> outside a finite exceptional <set>. Sorting that <set> gives a unique finite record of pairs $(x,f(x))$. <Real-number pairing by separated digits> encodes the record in $(0,1)$, and placing length-$n$ records in disjoint intervals $(n,n+1)$ gives an <injection> from all finite modifications into the real line. Conversely, changing the value at one fixed point gives a real-parameter family, so the <Cantor-Schröder-Bernstein theorem> shows this family has continuum cardinality.
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