  For one-dimensional interval and integrable weight function $w$ we define via
completion a weighted Sobolev space $H^{m,p}_{\mu_w}$ of arbitrary integer
order $m$. The weights in consideration may suffer strong degeneration so that,
in general, functions $u$ from $H^{m,p}_{\mu_w}$ do not have weak derivatives.
This contribution is focussed on studying the continuity properties of
functions $u$ at a chosen internal point $x_0$ to which we attribute a notion
of criticality of order $k$ and with respect to the weight $w$. For
non-critical points $x_0$ we formulate a local embedding result that guarantees
continuity of functions $u$ or their derivatives. Conversely, we employ duality
theory to show that criticality of $x_0$ furnishes a smooth approximation of
functions in $H^{m,p}_{\mu_w}$ admitting jump-type discontinuities at $x_0$.
The work concludes with demonstration of established results in the context of
variational problem in elasticity theory of beams with degenerate width
distribution.
