We discuss the analytic continuation of the Hadamard product of two holomorphic functions under assumptions pertaining to Ecalles Resurgence Theory, proving that if both factors are endlessly continuable with prescribed sets of singular points $A$ and $B$, then so is their Hadamard product with respect to the set ${0}cup A cdot B$. In this generalization of the classical Hadamard Theorem, all the branches of the multivalued analytic continuation of the Hadamard product are considered.
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