Definition of Time Invariance
A system $ \,s(t)\, $ is called time invariant if for any input signal $ \,x(t)\, $ yielding output signal $ \,y(t)\, $ and for any $ \,t_o\in\mathbb{R}\, $, the response to $ \,x(t-t_o)\, $ is $ \,y(t-t_o)\, $.
Example of a Time Invariant System
The following system is time invariant:
$ \,s(t)=2x(t-3)\, $
Proof:
Example of a Time Variant System
The following system is time variant:
$ \,s(t)=\, $
