Prove continuity of the algebraic operations on R, as follows:Use the metric d(a

Question

Prove continuity of the algebraic operations on R, as follows:Use the metric d(a , b) = |a - b| on R and the metric onR2 given by the equation ((x , y), (x0, y0)) = max {|x -x0|, |y - y0|}. (a) Show that addition is continuous. (b) Show that multiplication is continuous [hint: Given (x0, y0) and 0 < < 1 , let 3 = /(|(x0| +| y0| + 1 ) and not that (c) Show that the operation of taking reciprocals is acontinuous map from R - {0} to R. [hint: Show the inverse image ofthe interval (a,b) is open. Consider five cases, according as a andb are positive, negative, or zero.] (d) Show that the subtraction and quotient operations arecontinuous. *************PLEASE SHOW ALL WORK~~~PLEASE!!!**************** Prove continuity of the algebraic operations on R, as follows:Use the metric d(a , b) = |a - b| on R and the metric onR2 given by the equation rho((x , y), (x0, y0)) = max {|x -x0|, |y - y0|}. (a) Show that addition is continuous. [ hint: Given ?, let partial = e/2 and note that d(x+ y, x0 + y0) |x- x0| + |y - y0|.] (b) Show that multiplication is continuous [hint: Given (x0, y0) and 0

Explanation / Answer

b)let m:RxR->R be defined by (x,y)->xy. let U=(a,b) bean open set in R. m^-1(a)={(x,y)in R^2| xy=a} and m^-1(b)={(x,y)inR^2| xy=b}. These equations, xy=b and xy=a, are essentiallyhyperbolas with asymptotes at x=0 and y=0. They are as in part a)form boundaries of the set m^-1(a,b). in fact after thiseverything is exactly like part a; c) is the map x-> 1/x which is a hyperbola so part b)applies. d) subtraction is just like addition and division is just thecomposition of multiplication and reciprocal which were provedabove.

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