Let $(X,\|\cdot \|_{sup})$ be a Banach space, where $X=\{\phi\in C[0,1]: \phi(1)=0\}$. Now, let $\{T(t)\}$ be the family of left translations, i.e. $(T(t)\phi)(x)=\phi(x+t)\cdot \chi_{x+t\le 1}$ for $x\in[0,1]$ and $t\ge 0$. I want to prove that this family of operators is a strongly continuous semigroup and that the growth bound is $-\infty$.
I already proved that this set is a semigroup. So, in order to conclude that $\{T(t)\}$ is a $C_0$-semigroup I need to show that the function $t \mapsto T(t)\phi$ is continuous in $[0,1)$ with values in $X$, for each $\phi\in X$, however, I don't know how.
Secondly, I know that $\omega_0(T) := \inf\{\omega \in \mathbb{R} : \exists M = M_\omega \ge 1 \text{ such that } \|T(t)\| \le M_\omega e^{\omega t} \text{ for every } t \ge 0\}$, but how can I obtain $\omega_{T}=-\infty$. I tried to show that $\|T(t)\| \le M_\omega e^{\omega t}$ holds for all $\omega<0$.
How can I finish this proof? Any hints?