  Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map
with the specification property, and $\varphi: X \mapsto \IR$ a continuous
function. We consider the set of points for which the Birkhoff average of
$\varphi$ does not exist (which we call the irregular set for $\varphi$) and
show that this set is either empty or carries full topological pressure (in the
sense of Pesin and Pitskel). We formulate various equivalent natural conditions
on $\varphi$ that completely describe when the latter situation holds and give
examples of interesting systems to which our results apply but were not
previously known. As an application, we show that for a suspension flow over a
continuous map with specification, the irregular set carries full topological
entropy.
