Let $(A, d_a)$ and $(B, d_b)$ be two metric space such that:
-$A$ is bounded set of real valued functions, $d_a$ is the metric corresponding to the sup norm.
-$B$ is a set of real valued functions.
We have $\mu: A \to B$ a linear application, think of an integral for example.
Is it true that $\mu(A$) is bounded $\iff \mu $ is continuous ?