For all linear and many nonlinear operators between Fréchet spaces, several continuity results are established. Especially, a new closed graph theorem is given.The Hellinger-Toeplitz theorem says that if (X, X ) and (Y , Y ) are linear dual pairs of vector spaces and f : X → Y is a τ (X, X )-weak continuous linear operator where τ (X, X ) is the strongest compatible topology named Mackey topology, then f is τ (X, X )-τ (Y, Y ) continuous, e.g., if (X, · ), (Y, · ) are normed spaces and a linear f : X → Y is · -weak continuous, then f is · -· continuous [1, pp. 168-169].In general, the Hellinger-Toeplitz theorem cannot be extended to linear operators between Fréchet spaces because there exist Fréchet spaces such that each of them has the trivial dual {0}, e.g., L p (0, 1) for 0 < p < 1, the space M(0, 1) of measurable functions, etc. Fortunately, the closed graph theorem implies a Hellinger-Toeplitz type result for linear operators between Fréchet spaces [1, p. 68].In this paper we would like to establish a Hellinger-Toeplitz type result for a mapping family including all linear operators and much more nonlinear mappings between Fréchet spaces. Recently, R. Li and S. Zhong [2][3][4] have established a closed graph theorem which