  We study the conditions imposed on matter to produce a regular (non-singular)
interior of a class of spherically symmetric black holes in the $f(T)$
extension of teleparallel gravity. The class of black holes studied is
necessarily singular in general relativity. We derive a tetrad which is
compatible with the black hole interior and utilize this tetrad in the
gravitational equations of motion to study the black hole interior. It is shown
that in the case where the gravitational Lagrangian is expandable in a power
series $f(T)=T+\underset{n\neq 1}{\sum} b_{n}T^{n}$ that black holes can be
non-singular while respecting certain energy conditions in the matter fields.
Thus the black hole singularity may be removed and the gravitational equations
of motion can remain valid throughout the manifold. This is true as long as $n$
is positive, but is not true in the negative sector of the theory. Hence,
gravitational $f(T)$ Lagrangians which are Taylor expandable in powers of $T$
may yield regular black holes of this type. Although it is found that these
black holes can be rendered non-singular in $f(T)$ theory, we conjecture that a
mild singularity theorem holds in that the dominant energy condition is
violated in an arbitrarily small neighborhood of the general relativity
singular point if the corresponding $f(T)$ black hole is regular. The analytic
techniques here can also be applied to gravitational Lagrangians which are not
Laurent or Taylor expandable.
