Is $f(t) = e^{X(t)}$ continuous when matrix $X(t)$ is continuous?

Question

I think $f(t) = e^{X(t)}$ is continuous over $t \in \mathbb{R}$ when the complex matrix valued mapping $t\mapsto X(t)$ is continuous - regardless of whether $X$ is infinite or finite dimension. I tried to think of a proof, but without much success.

The hard part is the infinite-dimensional case, and I do not know what a proof could look like.