Please solve question 2 completely LTE7:29 PM 4VOLTE leO46%PROBLEMS1. (a) For any real number y, define a(x) = x + iy (so that a is a complex-valued function defined on R). Show that a is continuous. (Thisfollows immediately from a theorem in this chapter.) Show similarlythat B(y) = x + iy is continuous.(b) Let f be a continuous function defined on C. For fixed y, let g(x) =f(x + iy). Show that g is a continuous function (defined on R). Showsimilarly that h(y) = f(x + iy) is continuous. Hint: Use part (a).(a) Suppose that f is a continuous real-valued function defined on aclosed rectangle [a, b] × [c, d]. Prove that if f takes on the values f(z)2.25. Complex Functions 517and f(w) for z and w in [a, b]× [c, d], then f also takes all valuesbetween f(2) and f(w). Hint: Consider g(t) = f(tz + (1 – t)w) for tin [0, 1].*(b) If f is a continuous complex-valued function defined on [a, b] ×[c, d], the assertion in part (a) no longer makes any sense, since wecannot talk of complex numbers between f(2) and f(w). We mightconjecture that f takes on all values on the line segment between f(z)and f(w), but even this is false. Find an example which shows this.3. (a) Prove that if ao, . .. , an-1 are any complex numbers, then there are, žn (not necessarily distinct) such thatcomplex numbers z1,z" + an-12"-1+ · · · + ao =II (z - z:).i=1(b) Prove that if ao, . .., an-1 are real, then z" + an-12*=1 + • · · + aocan be written as a product of linear factors z + a and quadraticfactors z? + az + b all of whose coefficients are real. (Use Problem24-7.)4. In this problem we will consider only polynomials with real coefficients.Such a polynomial is called a sum of squares if it can be written ashi? + ··· + hn² for polynomials h; with real coefficients.(a) Prove that if f is a sum of squares, then f(x) 0 for all x.(b) Prove that if f and g are sums of squares, then so is f g.(c) Suppose that f(x) 0 for all x. Show that f is a sum of squares. Hint:First write f(x) = x*g(x), where g(x) + 0 for all x. Then k must beeven (why?), and g(x) 0 for all x. Now use Problem 3(b).5. (a) Let A be a set of complex numbers. A number z is called, as in the realcase, a limit point of the set A if for every (real) & 0, there is apoint a in A with |2 – al

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Please solve question 2 completely LTE7:29 PM 4VOLTE leO46%PROBLEMS1. (a) For any real number y, define a(x) = x + iy (so that a is a complex-valued function defined on R). Show that a is continuous. (Thisfollows immediately from a theorem in this chapter.) Show similarlythat B(y) = x + iy is continuous.(b) Let f be a continuous function defined on C. For fixed y, let g(x) =f(x + iy). Show that g is a continuous function (defined on R). Showsimilarly that h(y) = f(x + iy) is continuous. Hint: Use part (a).(a) Suppose that f is a continuous real-valued function defined on aclosed rectangle [a, b] × [c, d]. Prove that if f takes on the values f(z)2.25. Complex Functions 517and f(w) for z and w in [a, b]× [c, d], then f also takes all valuesbetween f(2) and f(w). Hint: Consider g(t) = f(tz + (1 – t)w) for tin [0, 1].*(b) If f is a continuous complex-valued function defined on [a, b] ×[c, d], the assertion in part (a) no longer makes any sense, since wecannot talk of complex numbers between f(2) and f(w). We mightconjecture that f takes on all values on the line segment between f(z)and f(w), but even this is false. Find an example which shows this.3. (a) Prove that if ao, . .. , an-1 are any complex numbers, then there are, žn (not necessarily distinct) such thatcomplex numbers z1,z&#34; + an-12&#34;-1+ · · · + ao =II (z - z:).i=1(b) Prove that if ao, . .., an-1 are real, then z&#34; + an-12*=1 + • · · + aocan be written as a product of linear factors z + a and quadraticfactors z? + az + b all of whose coefficients are real. (Use Problem24-7.)4. In this problem we will consider only polynomials with real coefficients.Such a polynomial is called a sum of squares if it can be written ashi? + ··· + hn² for polynomials h; with real coefficients.(a) Prove that if f is a sum of squares, then f(x) > 0 for all x.(b) Prove that if f and g are sums of squares, then so is f g.(c) Suppose that f(x) > 0 for all x. Show that f is a sum of squares. Hint:First write f(x) = x*g(x), where g(x) + 0 for all x. Then k must beeven (why?), and g(x) > 0 for all x. Now use Problem 3(b).5. (a) Let A be a set of complex numbers. A number z is called, as in the realcase, a limit point of the set A if for every (real) & > 0, there is apoint a in A with |2 – al < e but z # a. Prove the two-dimensionalversion of the Bolzano-Weierstrass Theorem: If A is an infinite subsetof {a, b] × [c, d], then A has a limit point in [a, b] × [c, d]. Hint:First divide [a, b] × [c, d] in half by a vertical line as in Figure 7(a).Since A is infinite, at least one half contains infinitely many points ofA. Divide this in half by a horizontal line, as in Figure 7(b). Continuein this way, alternately dividing by vertical and horizontal lines.(The two-dimensional bisection argument outlined in this hint is so

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