Instantaneous frequency vs fourier frequency
the frequency of a sinusoid is normally thought to be undefined when there is no sinusoid (i.e. when it is turned "OFF").

now, there are at least a couple of different methods of turning it OFF.  the most obvious is to effectively multiply the waveform output by zero to disconnect the sinusoid from the output.

however if the output of your signal generator is always connected (so there is no multiplication by zero) to the look-up table (LUT) that has the waveform values in it (and let's say that those waveform values follow a sinusoidal function), then another way to stop the output is to stop advancing position in the LUT.  that would be like setting the instantaneous frequency to zero.  but, unless you're watching out for this, it could get stuck on a DC value that is non-zero.  so there are a lot of functions, besides the zero function, that have a frequency of zero.  any DC function has instantaneous frequency of zero.