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For one-dimensional interval and integrable weight function $w$ we define viacompletion a weighted Sobolev space $H^{m,p}_{\mu_w}$ of arbitrary integerorder $m$. 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 offunctions $u$ at a chosen internal point $x_0$ to which we attribute a notionof criticality of order $k$ and with respect to the weight $w$. Fornon-critical points $x_0$ we formulate a local embedding result that guaranteescontinuity of functions $u$ or their derivatives. Conversely, we employ dualitytheory to show that criticality of $x_0$ furnishes a smooth approximation offunctions in $H^{m,p}_{\mu_w}$ admitting jump-type discontinuities at $x_0$.The work concludes with demonstration of established results in the context ofvariational problem in elasticity theory of beams with degenerate widthdistribution.

Higher order weighted Sobolev spaces on the real line for strongly degenerate weights. Application to variational problems in elasticity of beams