Example of equivalent metrics on the same set such that uniform continuity of some function is not preserved

Question

Give example of a set $X$ and two metrics $d_1,d_2$ on $X$ such that $(X,d_1)$ and $(X,d_2)$ are topologically equivalent but there exist a function $f:X \to X$ which is uniformly $d_1$ continuous but not uniformly $d_2$ continuous . What I know is that the metrics cannot be strongly equivalent . Please help . Thanks in advance

Answer 1

Let $X=(0,1)$, and let $d_1$ be the usual Euclidean metric on $X$. For $x,y\in X$ let $$d_2(x,y)=\left|\frac1x-\frac1y\right|\;.$$

• Verify that $d_2$ is a metric on $X$ and is topologically equivalent to $d_2$.
• Consider the function $f(x)=1-x$.