is bounded linear operator necessarily continuous?

An operator $C$ is bounded iff the set {$\|Cx\|:\|x\|\leq 1$} is bounded $\Leftrightarrow$ there is a $M<\infty:\|Cx\|\leq M\|x\|$ for every $x\in U$.

Let $ε>0$. If $x,y\in U:\|x-y\|<ε/M$, then $\|Cx-Cy\|\leq M\|x-y\|<ε$. Thus $C$ is not only continuous but uniformly continuous also.

So, a bounded operator is always continuous on norm-spaces. Banach space is a norm-space which is complete, thus things are not different there.

This property is unrelated to the completeness of the domain or range, but instead only to the linear nature of the operator. Yes, a linear operator (between normed spaces) is bounded if and only if it is continuous.

Added @Dimitris's answer prompted me to mention, beyond the fact that the implication on normed spaces indeed is an equivalence, that it's the converse which holds in the wider context of topological vector spaces, while the proposition mentioned here fails: there are bounded discontinuous linear operators, yet every continuous operator remains bounded.