Turn on Rebuild Surface, if you want a continuos face of the surface.
With the Closed Boundary off, select the edge to be extended. Ctrl+Data will allow multiple edges of the same surface to be extended. With the Closed Boundary on, select the boundary edge to be extended. Ctrl+Data will allow multiple boundaries to be selected of the same surface.
Enter a data point to accept the extension.
Left: Closed Boundary off | Right: Closed Boundary on