We construct discontinuous points of the Lyapunov exponent of quasiperiodic Shr\"odinger cocycles in Gevrey space $G^{s}$ with $s>2$. In contrast, the Lyapunov exponent has been proved to be continuous in $G^{s}$ with $s<2$ by Klein and Cheng-Ge-You-Zhou. This shows that $G^2$ is the transition space for the continuity of the Lyapunov exponent.