Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables4. Prove the following:a) A closed interval a 5 x 5 b is a bounded closed set.b) If v{(x) and y2(x) are continuous for a 5 x 5 b and yl(x) 5 y2(x) for a 5 x 5 b,then the set G: a 5 x 5 b, yl(x) c y 5 y2(x) is a bounded closed set. Is G a closedregion? Explain.5. Prove the Intermediate Value theorem for functions of one variable: Iff (x) is continuousfor a 5 x 5 b and f (a) < 0, f (b) > 0, then there exists an xo, a i xo < 6, such thatf (yo) = 0. [Hint: Let E be the set of x on the interval for which f (x) < 0 and let xo belub E.]6. Prove: If f is a real-valued function defined on E ~ then , f is continuous on E~ if andonly if for every open interval a < x 6 b the set of all P for which a < f (P) < b is open.7. Let G be a closed set in E ~ Let . Q be a point of EN not in G and let f (P) be the distancefrom an arbitrary point P of G to the point Q.a) Show that f is continuous on G.b) Show that f has a positive minimum at a point Po of G. Is the point Po unique?8. Let G and H be two closed sets in EN without common points. Show by example thatthe greatest lower bound (glb) of the distance between a point of G and a point of H canbe 0, but if G is bounded the glb is positive and is attained for some pair of points.Suggested ReferencesApostol, Tom M., Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nded. Reading, Mass.: Addison-Wesley, 1974.Buck, R. C., Advanced Calculus. 3rd ed. New York: McGraw-Hill, 1978.Coddington, Earl A., and Norman Levinson, Theory of Ordinary Differential Equations.New York: McGraw-Hill, 1955.Courant, Richard J., and Fritz John, Introduction to Calculus and Analysis, vol. 1, New York:Interscience, 1965 and vol. 2, New York: John Wiley & Sons, 1965.Gantmacher, F. R., Theory oj'Matrices, trans]. by K. A. Hirsch, 2 vols. New York: ChelseaPublishing Co., 1960.Kaplan, Wilfred, Maxima and Minima. New York: John Wiley & Sons, 1999.Rudin, W., Principles of Mathematics Analysis. 3rd ed. New York: McGraw-Hill, 1976.Struik, Dirk J., Lectures on Classical Differential Geometry, 2nd ed. Reading, Mass.:Addison-Wesley, 196 1.