Page "Heterodyne" Paragraph 30

So if two sine wave signals are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately, with no product terms.

So the function F must be nonlinear.

The only drawback to using a nonlinear component rather than a multiplier is that, in addition to the sum and difference frequencies, it produces other unwanted frequency components called harmonics which must be filtered from the output to leave the desired heterodyne frequency.