Let f be a functional on an open subset S of n. Then f is continuously differentiable (C1) if and only if each of the

Let f be a functional on an open subset S of „œn. Then f is continuously differentiable (C1) if and only if each of the partial derivatives Dif[x] exists and is continuous on S.
We now present some extensions and alternative forms of the mean value theorem which are useful in certain applications. Theorem 4.1 applies only to functionals and has no immediate counterpart for more general functions. Furthermore, since the point of evaluation x is unknown, the mean value theorem is often more usefully expressed as an inequality

which follows immediately from (9). This alternative form of the mean value theorem generalizes to functions between normed linear spaces.

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