Although white makes good shape in the corner, white is still not alive. There is not really a default continuation from this position though
One way to continue is the shoulder hit.
White can play a cut to move out to the center. In the late midgame this could be acceptable, though in the opening this would certainly be a mistake because black can make shape easily with a keima and can build too much territory on the left side as well.
The following sequence occured in a professional game between Tamura Yasuhisa (4 pro dan) and Ishii Senji (5 pro dan) in 1896.
Black also has the option to play more agressively with the following move.