The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts.

Let ("X", "d") be a metric space and "f" : "X" → "X" a continuous function. A point "x" in "X" is said to be Lyapunov stable, if,

We say that "x" is asymptotically stable if it belongs to the interior of its stable set, "i.e." if,

Barbalat's lemma and stability of time-varying systems
Definition for discrete-time systems
Example
Stability for linear state space models