Problems

Find the largest set you can in which the following functions are continuous.

38           Let Show that f is not continuous at (0,0).

39            Suppose f(x, y) is continuous at (a, b). Prove that g(x) = f(x, b) is continuous at x = a.

40            Prove that if f(x) and g(x) are continuous at x = a and if h(u, v) is continuous at ( f(a), g(a)), then k(x) = h(f(x), g(x)) is continuous at x = a.

41             Prove that if f(x, y) and g(x, y) are both continuous at (a, b) and if h(u, v) is continuous at (f(a, b), g(a, b)), then k(x,y) = h(f(x,y),g(x,y)) is continuous at (a, b). The notation

lim _{(x,y)→(a,b)} f(x, y) = L

means that whenever (x, y) is infinitely close to but not equal to (a, b), f(x, y) is infinitely close to L.

42            Evaluate

43            Evaluate

44            Evaluate

45             Evaluate

46            Show that does not exist.