Your proof is correct, and the result is that the system is time varying because the response to $x(t-t_0)$ does not in general equal a delayed response to the input $x(t)$. Of course, for the special case $t_0=2\pi k$ the delayed response to $x(t)$ equals the response to $x(t-t_0)$, but for time-invariance this equality must hold for any $t_0$.


Because the impulse response completely characterizes an LTI system. The reason is an arbitrary input $x(t)$ can be written as an infinite sum of time shifted and scaled Delta functions: $$x(t) = \int_{-\infty}^{+\infty}x(\tau)\delta(t-\tau)d\tau$$ or alternatively, using the convolution operator, $$x(t) = x(t)*\delta(t)$$ Properties of linearity and time ...


For a discrete-time system if the impulse response is IIR and right sided, then the ROC will be outside of the largest pole. Else if the impulse response is IIR and left sided, then the ROC will be inside of the smallest pole. For all FIR (finite inpulse response) systems, the ROC will be all $z$ except possibly zero and/or infinity. For the continuous-...

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