Banach algebra
Garth Dales and Robert M. Solovay proved in 1976 that Kaplansky's conjecture, namely that every algebra homomorphism from the Banach algebra "C(X)" (where "X" is some compact Hausdorff space) into any other Banach algebra must be continuous, is independent of ZFC. CH implies that for any infinite "X" there exists a discontinuous homomorphism into any Banach algebra.