Linear time- or shift-invariant systems don't change the frequency of sinusoids

If we input a sinuosid to a linear time- or shift-invariant system, the output is a sinusoid of the same frequency, in general with a different phase and a different amplitude.

What is remarkable is that no matter how complicated a system may be, when the input signal is a sinusoid, the output signal is also a sinusoid with the same frequency! In a sense, the complexity of the system wilts in the face of a sinusoidal input signal.

Proof (continuous-time)