Example can be found, for instance, in Boris Weisfeller's paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see  

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197662

His example of a discontinuous representation $\rho$ of $SO(n, {\mathbb R})$ to a semidirect product $H$ of $SO(n, {\mathbb R})$ with some ${\mathbb R}^N$, works for $SL(2, {\mathbb R})$ as well. Weisfeller's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups. Actually, Weisfeller's example is even more dramatic: The image of the compact group $SO(n)$ under $\rho$ is *dense* in $H$.