Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number $$ d = d(\lVert-\rVert, \lVert-\rVert') := \sup_{0 \neq v \in V}\big| \log\lVert v\rVert - \log\lVert v\rVert'\big| $$ is finite. This equips the set of all norms on $V$ with the structure of a complete metric space. You can think of $C := \exp(d)$ as the smallest real number $\ge 1$ such that $1/C \cdot \lVert-\rVert' \le \lVert-\rVert \le C \cdot \lVert-\rVert'$.

Now let $\{\lVert-\rVert_t\}_{t \in \mathbf{R}}$ be a family of norms on $V$, parametrized by the real numbers (I would also be interested in more general parameter spaces). Let us say that this is a *continuously varying family of norms* on $V$ if the map
$$
\mathbf{R} \times V \to \mathbf{R}_{\ge 0},
\qquad
(t, v) \mapsto \lVert v\rVert_t
$$
is continuous.

**Question:** Let $\{\lVert-\rVert_t\}_{t \in \mathbf{R}}$, $\{\lVert-\rVert'_t\}_{t \in \mathbf{R}}$ be two continuously varying families of norms on $V$. Will the function
$$
\mathbf{R} \to \mathbf{R}_{\ge 0},
\qquad
t \mapsto d(\lVert-\rVert_t, \lVert-\rVert'_t)
$$
be continuous?