Are the following functions Lipschitz continuous on the indicated sets Incidentally, all of these ex

Are the following functions Lipschitz continuous on the indicated sets? (Incidentally, all of these examples are scalar-valued functions; vector-valued functions would not pose any additional difficulties except for the need to examine more components.)

1/(1+2) on the interval [-1,1]?
On [-∞,-1]?
2√(1+x) on the interval [-1,1]?
On [-∞,-1]?
(2x + 1)^(1/3) on the interval [-1,1]?
On [-∞,-1]?
sin(x) on the interval [-1,1]?
On [-∞,-1]?
sin(1 - x) on the interval [-1,1]?
On [-∞,-1]?
x^2 + y^2 on the square {x |
On R^2?

Hint: Observe that the function is the composition of 1/(1+2) with the absolute value function. Show that the composition of two Lipschitz functions is Lipschitz continuous. Thus, x^2 + y^2 is Lipschitz on the indicated set if 1 + y^2 is.