We study the regularity of the Lyapunov exponent for quasi-periodic cocycles
$(T_\omega, A)$ where $T_\omega$ is an irrational rotation $x\to x+ 2\pi\omega$
on $\SS^1$ and $A\in {\cal C}^l(\SS^1, SL(2,\RR))$, $0\le l\le \infty$. For any
fixed $l=0, 1, 2,..., \infty$ and any fixed $\omega$ of bounded-type, we
construct a

$D_{l}\in {\cal C}^l(\SS^1, SL(2,\RR))$ such that the Lyapunov exponent is
not continuous at $(T_\omega, D_{l})$ in ${\cal C}^l$-topology.