When is the cohomology of the limit not the limit of the cohomology?

Let $G$ be a profinite group, and $A$ a G-module. If $G$ is the projective limit of {$G_{\alpha}$}, and $A$ the direct limit of {$A_{\alpha}$}, then
$H^*(G,A)$ is isomorphic to $dir lim_{\alpha} H^*(G_{\alpha},A_{\alpha})$.
Here the cohomology groups are defined via the group of continuous functions from $G$ to $A$. At page 26 of this book it is asserted that this theorem is false on discontinuous cochains. And my question is: is there an example illustrating this statement?
So you're asking for an example where $H^*_{\text{cont.}}(G,A) \not\cong H^*_{\text{discont.}}(G,A)$? –  Bruno Joyal Dec 4 '12 at 18:52
I mean this isomorphism, which I referred to earlier as the equality: $H^*_{\text{cont.}}(G,A) \cong dir limH^*_{\text{cont.}}(G_{\alpha},A_{\alpha})$ That is to say, the "theorem" alludes to that isomorphism exactly. –  awllower Dec 5 '12 at 14:43