Values of the scalar field c (x, t), if initially bounded, will always be bounded by the limits set by the initial conditions. This observation permits the maximum variance c'2 to be computed as a function of the mean value c. It is argued that this maximum should be expected in the limit of infinite Schmidt numbers (zero scalar species diffusivity). This suggests that c'/ c on the axis of turbulent jets, for example, may not tend to a constant, i.e., independent of x/d, in the limit of very large Schmidt numbers. It also underscores a difficulty with the k-1 scalar spectrum proposed by Batchelor [J. Fluid Mech. 5, 113 (1959)].