Let $B$ be a Banach space, $\epsilon > 0$, and $$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$
Denote $\partial/\partial t$ by $D_t$. Show that $(D_t t)$ is an isomorphism from $C_1^0 ([-\epsilon,\epsilon],B)$ onto $C^0([-\epsilon,\epsilon],B)$.
Note: I think I've already covered showing that $(D_t t)$ is a bijection. So my main problem is to show that $(D_t t)$ is a continuous operator.