The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Problems

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

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38           Let 11_partial_differentiation-152.gif 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 11_partial_differentiation-153.gif

43            Evaluate 11_partial_differentiation-154.gif

44            Evaluate 11_partial_differentiation-155.gif

45             Evaluate 11_partial_differentiation-156.gif

46            Show that 11_partial_differentiation-157.gif does not exist.


Last Update: 2006-11-25