We have a metric space $(X,d)$, functions $\phi, \phi_n : \left[0,\infty\right) \rightarrow \left[0,\infty\right)$, $n\in \mathbb{N}$. What does mean the sentence "$\phi_n \to \phi$ uniformly on the range of $d$ "?
I guess it means that $\phi_n \to \phi$ uniformly in $\left[0,\text{diam}\, X\right]$ if $\text{diam}\, X<\infty$ or in $\left[0,\infty\right)$ if $\text{diam}\, X=\infty$. Am I right?
It occurs here http://www.mathematik.tu-darmstadt.de/~kohlenbach/Briseid-phd.pdf, Definition 1.26). Thanks for a help.