  Given a henselian pair $(R, I)$ of commutative rings, we show that the
relative $K$-theory and relative topological cyclic homology with finite
coefficients are identified via the cyclotomic trace $K \to \mathrm{TC}$. This
yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity
theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's
theorem on relative $K$-theory (when $I$ is nilpotent).
  We deduce that the cyclotomic trace is an equivalence in large degrees
between $p$-adic $K$-theory and topological cyclic homology for a large class
of $p$-adic rings. In addition, we show that $K$-theory with finite
coefficients satisfies continuity for complete noetherian rings which are
$F$-finite modulo $p$. Our main new ingredient is a basic finiteness property
of $\mathrm{TC}$ with finite coefficients.
