Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote those $H^1$-functions that are also radial. Now, away from the origin, could we say that an element of $H^1_rad(X)$ is actually continuous? This seems to be akin to a Sobolev embedding type statement $H^1_{rad} (X \setminus K) \subset C(X \setminus K)$, where $K$ is a compact region containing the origin. My questions are: does the above inclusion hold in general, or is some kind of restriction on $\varphi(r)$ needed? Also, is the above inclusion continuous? Thanks.
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People do strange things from time to time. But if the question is reasonable (I'm also far enough away from the area that I'm agnostic, but of course, moderators have to be especially careful with close votes), then people can still benefit from it, and from reading its answers. $\endgroup$