2. Show that f(z)-İzl is continuous everywhere and differentiable nowhe how that f(z)lz is cont that...
1. Consider the function f(a) 2-9 3, but not differentiable at z Show that f( ) is continuous at r- 3.
Problem 2. Assume that f is differentiable everywhere. Prove that if is even, then f' is odd, and vice versa.
(4) Let f(x) (0 if x<0 (a) Show that f is differentiable at z (b) Is f'continuous on R? Is f continuous on R? Justify your answer.
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
3. (12) If the function below is continuous and differentiable everywhere within the interval [-3, 3), sketch the graph which satisfies the following conditions: f(-3) = 0, f(-2) = 3, f(-1) = 2, f(1) = -1, f(3) = -2 f'(-2)=0 f'(x) >0 on (-3,-2), f'(x) <0 on (-2, 3) f"(x) > 0 on (-1, 1), f '(x) <0 on (-3,-1) U(1, 3)
3. (12) If the function below is continuous and differentiable everywhere within the interval (-3, 3), sketch the graph which satisfies the following conditions: f(-3) = 0, f(-2) = 3, f(-1) = 2, f(1) = -1, S (3) = -2 f'(-2) = 0 f'(x) > 0 on (-3,-2), f '(x) <0 on (-2, 3) f"(x) > 0 on (-1, 1), f '(x) <0 on (-3,-1) (1, 3)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Find a holomorphic function F(z) on Ω-{z I Izl < r} such that for any a E Ω, F(a) F(0)-Z dz. Suppose f(z) is entire and Ω is simply connected domain. Show lim 22-h2220 Find a holomorphic function F(z) on Ω-{z I Izl
4. Let f(z) 22. (a) Show thatf is differentiable at z = 0. (b). Show that f is not differentiable at any point z # 0.
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g? (a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?