  We prove that a locally constant $SL_{2}(\mathbb{R})$-valued cocycle over the
shift generated by an irreducible collection of matrices is a continuity point
for Lyapunov exponents in the $\alpha$-H\"older topology for every $\alpha >
0$. This gives negative answers to conjectures of Viana and the author; we pose
a new conjecture to replace these conjectures. We show that an analogous
continuity result also holds for $GL_{2}(\mathbb{R})$-valued cocycles that
admit canonical holonomies.
