Is this piecewise-defined system time-invariant? \begin{equation}
  G_c \qquad y[n] =
    \begin{cases}
      u[n] &amp; \text{if} &amp; n&lt;0\\
      2u[n] &amp; \text{if} &amp; n\ge 0
    \end{cases}       
\end{equation}
I found this question in web: Whether or not is this time-invariant. The short answer is no. But I want to make sure I know how to prove this.
Let's take a certain $n&lt;0$ then $y[n] = 2u[n]$. Now, let's put instead of $n \rightarrow (n + n_0)&gt;0$. Since, it is time-invariant then $y[n+n_0] = u[n+n_0]$.
But $(n+n_0)&gt;0$, so $y[n+n_0] = 2u[n+n_0]$. So the only chance is: $u[n+n_0] = 0$, but since $n$ and $n_0$ where randomly chosen then, this system can be time-varying as long as always $u[n+n_0]=0 \rightarrow u[n] =0$ and $y[n] =0$ which doesn't hold as a system. So it is not time-varying. Does this makes sense?
Let us suppose that the system with input $x$ is defined by:
\begin{equation}
  y[n] =
    \begin{cases}
      x[n] &amp; \text{if} &amp; n&lt;0\\
      2x[n] &amp; \text{if} &amp; n\ge 0
    \end{cases}       
\end{equation}
As answered by @TimWescott, the two-part  definition suggests a lack of time invariance. Finding a counter-example could be a good start. For instance a two-part signal whose behavior when multiplied by 2 is visible. Indeed a modification of the Heavide jump is interesting:
\begin{equation}
  x_0[n] =
    \begin{cases}
      -1&amp; \text{if} &amp; n&lt;0\\
      \phantom{-}1 &amp; \text{if} &amp; n\ge 0
    \end{cases}       
\end{equation}
along with the right-shifted version:
\begin{equation}
  x_1[n] =
    \begin{cases}
      -1 &amp; \text{if} &amp; n&lt;1\\
      \phantom{-}1 &amp; \text{if} &amp; n\ge 1\,.
    \end{cases}       
\end{equation}
You get:
\begin{equation}
  y_0[n] =
    \begin{cases}
      -1&amp; \text{if} &amp; n&lt;0\\
      \phantom{-}2 &amp; \text{if} &amp; n\ge 0
    \end{cases}       
\end{equation}
along with the output for the right-shifted version:
\begin{equation}
  y_1[n] =
    \begin{cases}
      -1 &amp; \text{if} &amp; n&lt;0\\
      -2 &amp; \text{if} &amp; n=0\\
      \phantom{-}2 &amp; \text{if} &amp; n\ge 1\,.
    \end{cases}       
\end{equation}
which are not time-shifted versions of each other.