Melt butter with oil in a large skillet over medium-high heat; add cabbage, and sauté 10 minutes.
Stir in apple, and set aside.
Melt sugar in a large heavy saucepan over medium heat, stirring constantly; stir in vinegar (mixture will bubble, and sugar will clump), and bring to a boil.
Cook, stirring often, until sugar dissolves and mixture is caramel colored.
Stir in cabbage mixture; cook over medium heat, stirring occasionally, 20 minutes.