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In the theory of holomorphic functions of several variables, Hartogs's theorem that any holomorphic function on a punctured open set of \mathbb C^n http://latex.mathoverflow.net/png?%5Cmathbb%20C%5En ( n\geqslant 2 http://latex.mathoverflow.net/png?n%5Cgeqslant%202 ) can holomorphically be continued through the deleted point really started the subject and is still the most spectacular divide between n \geqslant 2 http://latex.mathoverflow.net/png?n%20%5Cgeqslant%202 and $n=1$" /> [ where it is completely false : on \mathbb C^\ast http://latex.mathoverflow.net/png?%5Cmathbb%20C%5E%2A look at 1/z or, worse, exp(1/z): these functions clearly can't be continued holomorphically through zero]