 Solution

If $f$ is a bounded linear operator, linearity gives $\Vert f(u)-f(v)\Vert \le C\Vert u-v\Vert$. It is therefore Lipschitz and continuous everywhere.

Conversely, suppose $f$ is continuous at zero. There is $\delta >0$ such that $\Vert u\Vert <\delta$ implies $\Vert f(u)\Vert <1$. For $u\ne 0$, apply this to $v=\delta u/(2\Vert u\Vert )$ and use homogeneity:The value at $u=0$ is zero by linearity. ConsequentlyThus boundedness and continuity are equivalent for linear maps between normed spaces. No finite dimensionality or completeness is needed for this equivalence.