Why does the author said "When $\mathcal{L}(E, F)$ is a Banach space, we can meaningfully speak of the continuity of the derivative"?

Question

I'm reading about the derivative in my analysis textbook:

I could not understand why the author said that

When $\mathcal{L}(E, F)$ is a Banach space, we can meaningfully speak of the continuity of the derivative.

IMHO, I usually do exercises about limits and continuity in metric spaces. On the other hand, $\mathcal{L}(E, F)$ is a metric space with the operator norm.

My question:

Is there any reason that the author said "When $\mathcal{L}(E, F)$ is a Banach space, we can meaningfully speak of the continuity of the derivative"?