continuity of homomorphisms on complete metrizable topological algebras

Let a be an f-algebra over the complex field. let b be an f-algebra such that the intersection of kernels of all continuous multiplicative linear functionals on b is singleton zero. if any nonempty open subset of the collection m(b) of all continuous multiplicative linear functionals in the gelfand topology contains uncountably many functionals or if b is a commutative frechet algebra such that m(b) has no isolated points, then any homomorphism from a onto a dense finitely generated subalgebra of b is continuous. this result has been proved in this article which is similar to a result derived by r.l. carpenter.