Let $X$ be a Banach space, $H\subseteq X$ be a dense hyperplane, and $f$ be a continuous linear functional defined on $H$. Then $f$ is uniformly continuous and hence it admits a unique continuous extension to $X$.
However, let us instead choose a discontinuous linear functional $g$ extending $f$ to the whole of $X$.
One cannot say that $g$ is too bad since, after all, its restriction to a big subspace, namely $H$, is continuous.
Does every Banach space admit a linear functional which is either zero or discontinuous when restricted to every infinite dimensional subspace?
The kernel of such a functional will then have a dense intersection with every infinite dimensional subspace.