  We study the pointwise perturbations of countable Markov maps with infinitely
many inverse branches and establish the following continuity theorem: Let $T_k$
and $T$ be expanding countable Markov maps such that the inverse branches of
$T_k$ converge pointwise to the inverse branches of $T$ as $k \to \infty$. Then
under suitable regularity assumptions on the maps $T_k$ and $T$ the following
limit exists: $$\lim_{k \to \infty} \dim_\mathrm{H} \{x : \theta_k'(x) \neq 0\}
= 1,$$ where $\theta_k$ is the topological conjugacy between $T_k$ and $T$ and
$\dim_\mathrm{H}$ stands for the Hausdorff dimension. This is in contrast with
the fact that other natural quantities measuring the singularity of $\theta_k$
fail to be continuous in this manner under pointwise convergence such as the
H\"older exponent of $\theta_k$ or the Hausdorff dimension $\dim_\mathrm{H}
(\mu \circ \theta_k)$ for the preimage of the absolutely continuous invariant
measure $\mu$ for $T$. As an application we obtain a perturbation theorem in
non-uniformly hyperbolic dynamics for conjugacies between intermittent
Manneville-Pomeau maps $x \mapsto x + x^{1+\alpha} \mod 1$ when varying the
parameter $\alpha$.
