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So I am confused on whether the statement that "a function f exists at all points in an open subset U of (say) R" , indicates that it is well-defined on all the points in that subset i.e will the function have a real value on all the points in the subset?

Also, can the derivative of a function exist at a point if the function is not well-defined at the point? For example, if the function goes to infinity at a point, is it possible for its derivative to exist at that point?

thanks!

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# Existence of a function vs being well-defined?

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