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Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions $z\backslash mapsto\; \backslash Re(z)$, $z\backslash mapsto\; and\; are\; not\; holomorphic\; anywhere\; on\; the\; complex\; plane,\; as\; can\; be\; shown\; by\; their\; failure\; to\; satisfy\; the\; Cauchy-Riemann\; conditions\; (see\; below).$

An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy-Riemann conditions.

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