CORRIGENDA TO “A SUFFICIENT CONDITION FOR A FUNCTION TO SATISFY
A WEAK LIPSCHITZ CONDITION”
by Radu Miculescu
MATHEMATICAL REPORTS9(59) (2007),3, 275–278
The correct form of the abstract is as follows.
Given a functionf :X → Y and two metric spaces (X, d) and (Y, ρ), consider the following condition: for anyx∈X there existsCx∈(0,1) such that for any sequenceS= (xn)n∈N∗, xn∈X,satisfying the conditiond(xn, x)≤Cxn,n∈N∗, there existsMS ∈ R+ such thatρ(f(xn), f(x))≤MSCxn, n ∈ N∗. It is shown that this condition is sufficient forf to be Lipschitz at each point ofX.
The correct form of the theorem on page 276 is as follows.
Theorem . Let (X, d) and (Y, ρ) be two metric spaces. Let f : X → Y be a function with the following property: for any x ∈ X there exists C
x∈ (0, 1) such that for any sequence S = (x
n)
n∈N∗, x
n∈ X, n ∈ N
∗, satisfying the condition d(x
n, x) ≤ C
xnfor all n ∈ N
∗, there exists M
S∈ R
+such that ρ (f (x
n) , f (x)) ≤ M
SC
xnfor all n ∈ N
∗. Then f is Lipschitz at any x ∈ X.
Finally, on page 276, line 7−, L should be M.
MATH. REPORTS10(60),3 (2008), 297
