Functional Analysis - Set of Applications is Open and Closed

Question

Let X,Y be two metric spaces, prove the following:

A) The set of bounded discontinuous applications is open in the set of the bounded applications from X to Y.

B)The set of bounded continuous applications is a closed subset of the set of bounded applications from X to Y.

Answer 1

The proof of $(B)$ is a straight generalisation of the Uniform Limit Theorem.

$(A)$ is equivalent to $(B)$.