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Is there a correct mathematical procedure to remove singular points so as to create a smooth continuum, differentiable everywhere? For example,for cusp singularities, is some kind of acceptable "cutting and joining" procedure available at the limit? I asked a similar question in the topology forum some time ago but never got an answer.

If we allow topological transformations, it seems to me that (for example) an inscribed equilateral triangle could be smoothly transformed to a circle without cutting and joining such that the points at the tips of the triangle become differentiable points on the circle.

http://mathworld.wolfram.com/SingularPoint.html

EDIT: I know "at the limit" can be problematical, but I'm trying to avoid arbitrary choices.

If we allow topological transformations, it seems to me that (for example) an inscribed equilateral triangle could be smoothly transformed to a circle without cutting and joining such that the points at the tips of the triangle become differentiable points on the circle.

http://mathworld.wolfram.com/SingularPoint.html

EDIT: I know "at the limit" can be problematical, but I'm trying to avoid arbitrary choices.

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