But can I say,'the random variable stays at C with probability 1'?
or 'if the characteristic function is 0, what is the probability function of the variable?'
Intriguing question. I suppose it depends on what "constant" means, when applied to a pdf.
The characteristic function of a random variable is the Fourier transform of the pdf of that random variable (perhaps a sign reversal in the exponent, depending on formulations).
Approaching this question like an EE and not a statistician,:
If by "constant" you mean "constant pdf," then this would describe a flat probability density function. So that's analogous to a single sine wave. A continuous function in the "time domain" is expected to result in a narrow frequency domain spectrum. And vice versa.
The Fourier transform would therefore be a delta function (impulse, infinite amplitude, no bandwidth), at the frequency of that time domain signal.
If the pdf is zero, however, then it seems inescapable that the Fourier transform is zero as well.
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