Discontinuity of Lyapunov exponents for SL$(2,\mathbb{R})$ valued cocycles

We exhibit an example of discontinuity point for the Lyapunov exponents as a function of the cocycle in the $\alpha$-H\"older topology. The linear cocycle taking values in $SL(2, \mathbb{R})$ is locally constant and defined over a Bernoulli shift. Our example extends Bocker-Viana's and Butler's results. In particular, it gives a partial answer to a question raised by Clark Butler. Finally, we give an example of discontinuity in the setting of $SL(m,\mathbb{R})$-valued cocycles, which is constructed from $SL(2, \mathbb{R})$-valued cocycles.