  We study the Dirichlet problem for least gradient functions for domains in
metric spaces equipped with a doubling measure and supporting a
(1,1)-Poincar\'e inequality when the boundary of the domain satisfies a
positive mean curvature condition. In this setting, it was shown by Mal\'y,
Lahti, Shanmugalingam, and Speight that solutions exist for continuous boundary
data. We extend these results, showing existence of solutions for boundary data
that is approximable from above and below by continuous functions. We also show
that for each $f\in L^1(\partial\Omega),$ there is a least gradient function in
$\Omega$ whose trace agrees with $f$ at points of continuity of $f$, and so we
obtain existence of solutions for boundary data which is continuous almost
everywhere. This is in contrast to a result of Spradlin and Tamasan, who
constructed an $L^1$-function on the unit circle which has no least gradient
solution in the unit disk in $\mathbb{R}^2.$ Modifying the example of Spradlin
and Tamasan, we show that the space of solvable $L^1$-functions on the unit
circle is non-linear, even though the unit disk satisfies the positive mean
curvature condition.
