Is there a strongly continuous one-parameter semigroup of operators $(T(t))_{t\geq 0}$ such that the map $M:[0,\infty)\times X\to X$ defined by $M:(t,x)\mapsto T(t)x$ is not continuous? where $X$ is a Banach space.
I notice that such a map $M$ is continuous if the semigroup $T(t)$ is continuous in the operator norm.
No, there isn't; the mapping $M$ is continuous for any strongly continuous one-parameter semigroup.
Hint: Fix $(t,x) \in [0,\infty) \times X$, then
$$|T(t)x-T(s)y| \leq |T(t)x-T(s)x| + |T(s)x-T(s)y| \leq c |T(|t-s|)x-x| + c|x-y| $$
for any $y \in X$, $s \in [0,2t]$ where $$c := \sup_{r \in [0,2t]} \|T(r)\|<\infty.$$