Given a continuous function f:[a, b] -> R that is differentiable on (a, b), the Mean-Value
Theorem (MVT) of elementary calculus allows us to 'predict" a value of f'(x).
Knowing only the boundary values f(a) and f(b), we predict that f'((a,b))э(f(b)-f(a))/(b-a). May we not expect an analogous result for f:K -> R when K is a.