The question is in the title really, but I suppose I could at least fix some notation here.
Let $X$ be an infinite dimensional Banach space - over the reals for the sake of concreteness. Use choice to produce a Hamel basis $(x_i)_{i \in I}$ for $X$. I have the impression that this basis should necessarily interact badly with the topology and, therefore, not be of much (any?) use for doing analysis on $X$, but I've never really thought about why this should be the case.
Is there a rigorous sense in which we can say that this basis is "useless"?
One way to make this precise (although I'd be interested in other points of view) might be to consider the corresponding linear functionals $(f_i)_{i \in I}$ - uniquely determined by taking $f_i(x_j)$ equal to $1$ or $0$ according as $i=j$ or not. I would be very surprised if it were possible for any of these functionals to be continuous, but this conviction is based only on my vague notion that a Hamel basis for $X$ must be somehow "pathological" and not on any sound reasoning.
Because I have no idea how to approach this question (admittedly also because I don't think this is a particularly constructive thing to be dwelling on at this time of year...) I thought I would appeal to the hard-earned wisdom of the good people of Mathematics Stack Exchange.
Edit for clarity: I actually already know at most finitely many of the coordinate functions can be continuous (see comment below) which is, now that I think about it, a fairly critical failure in and of itself and probably enough to justify the word "useless" above. My question is whether they must all be discontinuous.