Let $(M,d)$ be a metric space and $f\colon[0,\infty)\to[0,\infty)$ metric preseving map that is right continuous at $0$, i.e. $f$ has satisfies $$\forall x,y\in [0,\infty)\colon f(x+y)\le f(x)+f(y)\quad\text{and}\quad f(x)=0\iff x=0.$$ Furthermore, $f$ is non-decreasing and addition $$f(0^+)=\lim_{x\downarrow0}f(x)=f(0).$$
Then the composition $\Delta=f\circ d$ is again a metric on $M$. But how can I show $$(M,d) \text{ is complete}\iff (M,\Delta) \text{ is complete}?$$