The textbook says that this function has to be continuous at least in the origin for it to be continuous everywhere. But how is it possible that a function is already linear but somehow not continuous?
For example, $E$ and $F$ are two normed vector spaces. $f:E\rightarrow F$ is a linear function. Obviously we know that $f(0) = 0$. Now, for a non-zero vector $a$ in $E$, as long as $f(a)$ has a definition, say $b=f(a)$ for some $b\in F$. Then for however small $\epsilon$, as long as $\lVert x\rVert<\lVert a\rVert\frac{\epsilon}{\lVert b\rVert}$, we have $\lVert f(x)\rVert<\epsilon$. So it seems that this function is continuous at the origin without stipulating it.