I've got problem with following: $\phi : [0,\infty) \to \mathbb{R}$ is non-decreasing, concave function. Such that $\phi (0) =0$, and $\phi (u) >0$ for $u>0$. Prove that if $\phi$ is continuous at $0$ then $\mathcal{T} (d_{\phi}))=\mathcal{T} (d))$, where $d$ is a metric on $X$ and $d_{\phi}(x,y) = \phi (d(x,y))$.
I can prove that $d_\phi$ is metric on $X$. I've stucked with above problem.