A. correct. Prove it by reducing to absurdity. Please let me know if you need detailed proof.

B. no For example, y=sin( 1/x) is a bounded function, but when x approaches 0, the function has no limit, which means that the bounded function may have no limit.

C. correct. On the other hand, if the function is continuous, it may not be differentiable, for example, y=| x |.

The rest of the questions seem incomplete.