The aim of this paper is to solve a linear preserver problem on the fu
nction algebra C(X). We show that in the case in which X is a first co
untable 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 e
lement 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.