ULUM turu Problem 1. Assume that f:R R is continuous and satisfies f(x) - f(y) =...

Question

Question

Answer 1

Answer #1

Page NO.: A esume that f: TR TR is continous and satisfies Tflow) – f(y) = (x-y j for all X,Y E TR then to show f u constant.

Answer 2

Similar Homework Help Questions

Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for...

Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all x, y ∈ R. Prove that for some constant a ∈ R, f(x) = ax. Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all X, Y E R. Prove that for some constant a ER, f(x) = ax.
Show that is not uniformly continuous on . f:R +R, f(1) = x + 2.0 We...

Show that is not uniformly continuous on . f:R +R, f(1) = x + 2.0 We were unable to transcribe this image
1 EL CIUSCUJUDUCU UL . sin(1/2), 220 Problem 6: (10 pts) Let f:R → R be...

1 EL CIUSCUJUDUCU UL . sin(1/2), 220 Problem 6: (10 pts) Let f:R → R be defined by f(0) = x=0 Show that f'(2) exists for all ER, but the function f':R → R is not continuous at 0.

Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for...

Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all X,Y E R. Prove that for some constant a ER, f(x) = ar.
Problem 4 Let S :R R be such that f (x + y) = f(x) +...

Problem 4 Let S :R R be such that f (x + y) = f(x) + f(y) for all sy ER Also assume that limf () = LER. 1. Show that f (2x) = 2 (s). 2. Use the result from part 1 to determine the value of L.
Consider the function f:R + R defined by if x is rational f(x) = if x...

Consider the function f:R + R defined by if x is rational f(x) = if x is irrational. Find all c € R at which f is continuous. C

1. Let f:R → R be the function defined as: 32 0 if x is rational...

1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
(1) If f: R₃ R a continuous function such that f(x)² > 0 for all xER....

(1) If f: R₃ R a continuous function such that f(x)² > 0 for all xER. Show that either f(x) >0 for all a ER or f(x) <0 all X E R.
b) i. Using e-8 definition show that f is continuous at (0,0), where f(x,y) = {aš...

b) i. Using e-8 definition show that f is continuous at (0,0), where f(x,y) = {aš sin () + yś sin () if xy + 0 242ADES if xy = 0 ii. Prove that every linear transformation T:R" - R" is continuous on R". iii. Let f:R" → R and a ER" Define Dis (a), the i-th partial derivative of f at a, 1 sisn. Determine whether the partial derivatives of f exist at (0,0) for the following function. In...

Let f : R2 → R be a uniformly continuous function and assume that If(y,t)| M. Let yo E R. The goal of this exercise...

Let f : R2 → R be a uniformly continuous function and assume that If(y,t)| M. Let yo E R. The goal of this exercise is to show the existence of a function φ : [0, 1] → R that solves the initial value problem o'(t)-F(d(t),t), ф(0)-Yo (a) Show that there is a function n1,R that satisfies t <0 n(リーレ0+.GF(du(s-1/n),s)ds, t20. Hint: Define фп first on [-1,0] , then define фп。n [0,1 /n), then on [1/n, 2/n], and so on...