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We prove the existence of a universal Cauchy–Riemann function on   ℂ  n   ×   E   , where    E  ⊂ ℝ  mhas a regular exhaustion by compact sets. The proof requires first proving a Weierstrass-type approximation theorem for CR-functions on   ℂ  n   ×   E   , where    E  ⊂ ℝ  m    . The technique used in this latter theorem leads to a Runge-type approximation theorem for Cauchy–Riemann functions on a Cartesian product of compact subsets in   ℂ × ℝ  m    .

A universal Cauchy–Riemann function on subsets of ℂ n × ℝ m