In case \(A\) is a symmetric banded matrix and \(f\) a smooth function defined on an interval containing the spectrum of \(A\) it is shown that the entries of \(f(A)\) are bounded in an exponentially decaying manner away from the main diagonal. This result is a generalization of a result by S. Demko, W. F. Moss and P. W. Smith on the decay of the inverse [Math. Computer 43, 491-499 (1984; Zbl 0568.15003)]. If \(f(A)\) is represented by means of Riemann-Stieltjes integrals bounds can be obtained. This is also done in case that the integrals are approximated by Gaussian quadrature rules. Numerical examples illustrating the application of the obtained bounds to preconditioning are given.