As the Fed's current vice-chairman, she implies continuity in monetary policy.
We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as semismoothness.
The continuity of the operator F follows from the continuity of f, g, (I_{ik}) and (J_{ik}).
Theorem 3.3 can be proved without assuming the b-continuity of f or the b-continuity of g.
In particular, if F satisfies the Lipschitz continuity condition | F ( t, s, x ) − F ( t, s, y ) | ≤ | x − y |, t, s ∈ [ 0, 1 ], x, y ∈ R, (3.5).
Then lim T → ∞ P { H T ≤ z } = F ( z ) at every continuity point of F ( z ).
Obviously, the function f ¯ is continuous on I × R + × R − according to the continuity of f.
Cauchy's definition of continuity of f(x) in the neighbourhood of a value a amounts to the condition, in modern notation, that limx→af(x) = f(a).
So by continuity, F ( x n ) → F ( u ) as n → ∞ in S, and therefore u = F ( u ).
Remark 3.16 We note that in Theorem 3.15 the assumption that f is orbitally G-continuous can be replaced by orbital continuity or continuity of f.
The continuity of f implies that A is continuous.
