Diameter preserving linear bijections of C(X)

The aim of this paper is to solve a linear preserver problem on the function algebra C(X). We show that in the case in which X is a first countable compact Hausdorff space, every linear bijection phi : C(X) --> C(X) having the property that diam(phi(f)(X)) = diam(f(X)) (f is an element of C(X)) is of the form phi(f) = tau . f o phi + t(f)1 (f is an element of C(X)) where tau is an element of C, \tau\ = 1, phi : X --> X is a homeomorphism and t:C(X) --> C is a linear functional.