(a) Show that causality for a continuous-time linear system is equivalent to the following statement:
For any time t0 and any input x(t) such that x(t) = 0 for t < t0, the corresponding output y(t) must also be zero for t < t0.
The analogous statement can be made for a discrete-time linear system.
(b) Find a nonlinear system that satisfies the foregoing condition but is not causal.
(c) Find a nonlinear system that is causal but does not satisfy the condition.
(d) Show that invertibility for a discrete-time linear system is equivalent to the following statement:
The only input that produces y[n] = 0 for all n is x[n] = 0 for all n.
The analogous statement is also true for a continuous-time linear system.
(e) Find a nonlinear system that satisfies the condition of part (d) but is not invertible.
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