The following result is obtained.
Let f be a self map on a nonempty complete metric space (X,d). Then the following
conditions are equivalent: (i) For any 𝜖 > 0, there exists δ(𝜖) > 0 such that
d(f(x),f(y)) < 𝜖 whenever 𝜖 ≦ d(x,y) < 𝜖 + δ(𝜖). (ii) There exists a function w of
[0,∞) into [0,∞) such that w(s) > s for all s > 0, w is lower semicontinuous from the
right on (0,∞) and w(d(f(x),f(y))) ≦ d(x,y),x,y ∈ X.
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