The concept Open Neighborhood in a metric space (S^N, d) is reviewed:
- Open Neighborhood For all p > 0, and any s_0 \in S^N, the neighbor N(s_0, p) is defined as the set N(s_0, p) = {s: d(s, s_0) < p }.
There are two equivalent and general (in the sense of topoloical space) definitions of the concept Continuity:
- Definition A A function f defined on the topological space S is continuous iff, for all sequence in S s.t. {s_1, s_2, s_3,...} --> s ==> {f(s_1),f(s_2),f(s_3),... } --> f(s)\].
- Definition B A function f defined between the two topological spaces X and Y f: X --> Y is continuous iff for all open subset O of Y, f^{-1}(O) is open.
In the case of metric space (S,d), there is another way to define continuity:
- Definition C A function f is continuous if given any f(s_1) and any p>0, there exists q>0 s.t. For all s_2 in N(s_1,q) ==> f(s_2) is in N(f(s_1),p)
Especially, we have the concept of Lipschitz Continuous :
- Definition D f is Lipschitz continuous if there exists L >= 0 s.t. For all s, and s' in S we have d(f(s),f(s')) <= L*d(s,s').
Obviously, in a metric space, if a function f is causal, it is Lipschitz continuous as well. Another property of Lipschitz continous is:
Fact In a metric space, Lipschitz continous ==> continous.
Proof Given p > 0 and f(s_1), let q = p/L. Then if s_2 is in N(s_1,q) i.e. d(s_1,s_2) <= p/L, we have d(f(s_1),f(s_2)) <= L*p/L = p i.e. f(s_2) is in N(f(s_1),p).Note!! the other direction is not true: just think about the exponential function exp(x).