A system is called time invariant if for any input signal x(t)(x[n]) and for any t0 belongs to R, the response to the shifted inputX(t-t0) is the shifted output y(t-t0).
Example: X(t) ->SYSTEM -> y(t) = 10 x(t) is time invariant because X(t) -> t0 -> y(t) = X(t-t0) -> SYSTEM -> z(t) = 10 y(t) = 10 x(t-t0)
where as a system is called time variant when we find an input signal for which the condition of time invariance is violated. Example:
y[n] = nx[n] Proof: consider an input signal x1[n] = d[n] which yields an output y1[n] that is identically 0.However the input x2[n] = d[n-1] yields the output y2[n] = nd[n-1] = d[n-1].
