  Singular currents typically appear on rational surfaces in non-axisymmetric
ideal magnetohydrodynamic equilibria with a continuum of nested flux surfaces
and a continuous rotational transform. These currents have two components: a
surface current (Dirac $\delta$-function in flux surface labeling) that
prevents the formation of magnetic islands and an algebraically divergent
Pfirsch--Schl\"uter current density when a pressure gradient is present across
the rational surface. At flux surfaces adjacent to the rational surface, the
traditional treatment gives the Pfirsch--Schl\"uter current density scaling as
$J\sim1/\Delta\iota$, where $\Delta\iota$ is the difference of the rotational
transform relative to the rational surface. If the distance $s$ between flux
surfaces is proportional to $\Delta\iota$, the scaling relation
$J\sim1/\Delta\iota\sim1/s$ will lead to a paradox that the Pfirsch--Schl\"uter
current is not integrable. In this work, we investigate this issue by
considering the pressure-gradient-driven singular current in the
Hahm\textendash Kulsrud\textendash Taylor problem, which is a prototype for
singular currents arising from resonant magnetic perturbations. We show that
not only the Pfirsch--Schl\"uter current density but also the diamagnetic
current density are divergent as $\sim1/\Delta\iota$. However, due to the
formation of a Dirac $\delta$-function current sheet at the rational surface,
the neighboring flux surfaces are strongly packed with
$s\sim(\Delta\iota)^{2}$. Consequently, the singular current density
$J\sim1/\sqrt{s}$, making the total current finite, thus resolving the paradox.
