2. Suppose f is holomorphic in an open set UC C and a € U. Show that for any integer n ≥ 0 there is a unique holomorphic function h : U → C such that ƒ(z) = f(a)+ƒ'(a)(z−a)+ƒ"(a). (z − a)² 2! (z − a)n n! ·+...+ f(n) (a). -+(z−a)"+¹h(z), z EU.

Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) < f2(0) and f1(1) > f2(1). Show that there exists a point c ∈ (0,1) such that f1(c) − f2(c) = 0.

4. Let f: [-1, 1] → R be a continuous function. Prove the following statements: (a) If there is c € [-1,1] such that f(c)f(−c) < 0, then there is de R such that f(d) = 0. (b) If ƒ([-1, 1]) = (-1, 1), then f is not continuous.

Let f be a function defined on all of R that satisfies theadditive condition f(x + y) = f(x) + f(y) for all x, y ∈ R (a) Show that f(0) = 0 and that f(−x) = −f(x) for all x ∈ R. (b) Let k = f(1). Show that f(n) = kn for all n ∈ N, and then prove thatf(z) = kz for all z ∈ Z. Now, prove that f(r) = kr for any rationalnumber r.