Finite modification of the identity on the real line ($|\{x:f(x)\ne x\}|<\infty$)

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.