For $x,y\in X$, the <triangle inequality> and the fact that $\phi$ is an <isometry> give
$$
\begin{aligned}
|d_\phi(x)-d_\phi(y)|
&=|d(x,\phi x)-d(y,\phi y)|\\
&\leq d(x,y)+d(\phi x,\phi y)\\
&=2d(x,y).
\end{aligned}
$$
Thus the <displacement function> is $2$-Lipschitz, hence <continuous function>[continuous].

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 133 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-133/4/a/solution