Identity theorem  In complex analysis, a branch of mathematics, the identity theorem for holomorphic functions states: given functions "f" and "g" holomorphic on a domain "D" (open and connected subset), if "f" = "g" on some formula_1, formula_2 having an accumulation point, then "f" = "g" on "D". Thus a holomorphic function is completely determined by its values on a single open neighborhood in "D", or even a countable subset of "D" (provided this contains a converging sequence). This is not true for real-differentiable functions. In comparison, holomorphy, or complex-differentiability, is a much more rigid notion. Informally, one sometimes summarizes the theorem by saying holomorphic functions are "hard" (as opposed to, say, continuous functions which are "soft"). The underpinning fact from which the theorem is established is the expandability of a holomorphic function into its Taylor series. The connectedness assumption on the domain "D" is necessary. For example, if "D" consists of two disjoint open set, formula_3 can be formula_4 on one open set, and formula_5 on another, while formula_6 is formula_4 on one, and formula_8 on another. If two holomorphic functions "f" and "g" on a domain "D" agree on a set S which has an accumulation point "c" in "D", then "f = g" on a disk in formula_9 centered at formula_10. To prove this, it is enough to show that formula_11 for all formula_12. If this is not the case, let "m" be the smallest nonnegative integer with formula_13. By holomorphy, we have the following Taylor series representation in some open neighborhood U of "c":  By continuity, "h" is non-zero in some small open disk "B" around "c". But then "f" − "g" ≠ 0 on the punctured set "B" − {"c"}. This contradicts the assumption that "c" is an accumulation point of {"f = g"}. This lemma shows that for a complex number "a", the fiber "f"−1("a") is a discrete (and therefore countable) set, unless "f" ≡ "a". Define the set on which formula_3 and formula_6 have the same Taylor expansion:  We'll show formula_18 is nonempty, open, and closed.