Let f be a function defined on an interval I. All we know about f is that there is a constant K> 0 so that | f (a) - f (b) | ≤ K | a - b | ^ 2 ,for all a, b ∈ I. Show that f must be constant on I.

Let f be a function defined on an interval I. All we know about f is that there is a constant K> 0 so that | f (a) - f (b) | ≤ K | a - b | ^ 2 ,for all a, b ∈ I. Show that f must be constant on I.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.1: Polynomial Functions Of Degree Greater Than

Problem 36E

See similar textbooks

Related questions

Q: Suppose f and g are non-constant, differentiable, realvalued functions defined on (−∞, ∞).…

A: We plug x=0 and y=0

Q: Prove that f(x) = x 3 - 12x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Since, we know that a polynomial function is always differentiable.⇒f(x)=x3-12x-(i) is always…

Q: The function (f/g)(x) is defined for all values of x that are in the domain(s) of both f and g, and…

A: The denominator can not be equal to 0 Because any constant divided by 0 is undefined

Q: Suppose f is a function which satisfies the properties: f(x + y) = f(x) + f(y) + x²y + xy? for all…

A: Given that, f(x+y)=f(x)+f(y)+x2y+xy2 for all real numbers 'x' and 'y' and, limx→0f(x)x=1 As we…

Q: Let f(x) be a function that is differentiable on the entire real line. If f( – 6) = 8 and f(9) = 8,…

A: Given f-6=8, f9=8

Q: Let f denote the function if x < 2; 1 f(x) = if x 2. Which of the following is true about f? Of is…

A: Given f(x)=x24if x≤22xif x>2

Q: Let f be defined in an open interval that contains a=3 and assume that all x,…

A: f be defined in an open interval that contains a=3 and assume that all x,2x+6≤f(x)≤x2-4x+15

Q: A function f is said to have a removable discontinuity at (a) if:1. f is either not defined or not…

A: We have to factor the numerator At x=3 both of the numerator and denominator are zero, so we get…

Q: 1. Let f be defined on an interval I and let x1 and x2 be numbers on I. If f(x1) < f(x2) whenever x1…

A: Given that: f is deifned in I such that: f(x1)<f(x2) whenever x1<x2

Q: Find the value of c such that the function is continuous on the entire real number line. [x + 3, xs2…

Q: (a) Assume that f(x) is a function defined by f(x) = x²-3x+1 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x)…

A: Given.

Q: A function f is ___________ on an interval I if, for any choice of x1 and x2 in I, with x1 < x2,…

A: To fill: The blank of the statement. Given: The statement is "A function f is _______ on an open…

Q: : Let f [a, b] → R be a continuous and one-to-one function, where a < b are real numbers. Use the…

A: Introduction: There must be at least one "c" such that c∈(a,b) (or a<c<b) and f(c) = L. if…

Q: Determine whether there exists a function f : [0, 2] → R or none such that f(0) = -1, f(2)= 4 and…

A: In this question, we need to determine whether there exists any function for the given conditions.

Q: Prove that if (c, f(c)) is a point of inflection of the graph of f and f" exists in an open interval…

A: It is given that (c,f(c)) is a point of inflection of the function f.

Q: Verify the mean value theorem on the given interval, and find all values of c in that interval that…

Q: Let f be a real-valued function continuous at the point a = (a1, a2) in R². Keep a2 fixe and define…

A: Let f be a real-valued function continuous at the point a=(a1,a2). Define g as g(x)=f(x,a2), where…

Q: Suppose f and g are contimuously differentiable functions such that f(x) = d(x) and g(x) f(x) and…

A: Define a new function r(x) = f2(x) -…

Q: A function f() is said to have a removable discontinuity at z = a if: 1. f is either not defined or…

A: A function fx has a removable discountinuity at x=a if limx→a fx=limx→a1 fx ≠fa

Q: Prove that if f (0) = g(0) and f' (x) ≤ g' (x) for x ≥ 0, then f (x) ≤ g(x) for all x ≥ 0. Hint:…

Q: Let f be the function defined above, where b is a constant. For what values of b, if any, is f…

A: Left limit at 3 should be equal to right limit at 3 for the function to be continuous.

Q: Suppose we have a function f on an interval [a,b] where f(a)=f(b)=0 and f''(x) <= 0 for all x on the…

Q: Does there exist a function f such that f(0) =-1, f(2) =4, and f'(x) ≤ 2 for all x?

Q: Assume that f(x) is a function defined by x²-3x+1 f(x) = 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x) is…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Prove that f(x) = (x- 1)2 + (x + 1)? is an even function. A function k(x) is an even function if…

Q: Use Rolle's Theorem to prove that if f'(x) = 0 for all x in an interval ( a , b ), then f is…

A: Using Rolle's Theorem to prove that if f'(x) = 0 for all x in an interval ( a , b ), then f is…

Q: Let f (x) = 2x3 - 3x2 - 36x + 14. Also, let a = 0 and b = 1. (a) Show that the function f is…

Q: A function f(x) is said to have a removable discontinuity at x = a if: 1. f is either not defined or…

A: Here is the solution:

Q: Let f be the function given by f(x)=x3. What are all values of c that satisfy the conclusion of the…

Q: show that any function f(x) defined on a symmetrically placed interval can be written as the sum of…

A: Here we have given one function, and we have to prove that the function f(x) can be written as the…

Q: Define h as x 2 Determine whether the function h is continuous h(x) = {3, x2 or discontinuous at x…

A: NOTE: Refresh your page if you can't see any equations. . if the function h(x) is continuous at x=2…

Q: Prove if f and g are both functions that are continuous at some point a, then the difference f - g…

Q: Let f be the function defined by f (x) = 6x. On which open intervals is ƒ decreasing? %3D -2 3 <글…

A: Given that: f(x)=x33-x22-6x

Q: 13. Prove: If a function f is differentiable at some a in its domain, then f is also continuous at…

A: We have a property that, "Every differentiable function is continuous but every continuous function…

Q: Find the constant a such that the function is continuous on the entire real number line. x < 1 x 2 1…

A: The function will be continuous when Left hand limit= Right hand limit LHL=RHL

Q: If f(a)> f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a…

A: If f(a)> f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a relative…

Q: Let f be a function defined by f(x) = In(2x2 + 1) and let g be a function defined by g(x) = 2x4 –…

Q: Let n be a positive integer and let f: [0..n] - (0..n] be an injective function. Define the function…

Q: Let us define a number a to be a fixed point of a function f if a = f(a). (For example, if f (x) =…

A: We have to prove that a function has at most one fixed point if f'x≠1 Let us assume that the…

Q: Let [a,b] be a closed and bounded interval, let keR, and let f:[a,b]¬R be a function. Suppose that f…

Q: Suppose f is a function which satisfies the properties: f(x + y) = f(x) + f(y) + x²y + ry? %3D for…

A: It is given that fx+y=fx+fy+x2y+xy2 for all real numbers x and y. Substitute y=0 in…

Q: Let f be a function on an interval I and let ₁ and 2 be any two points in I. Then if f (x₁) > f (x₂)…

Q: Determine if the function f(x) = 6/1 – x² satisfies the Mean Value Theorem on [0, 1]. If so, find…

A: givenf(x)=61-x2interval is [0,1]according to mean value theorem f'(c)=f(1)-f(0)1-0

Q: Let f be the function Зх + 5 for x 5 х Compute the left- and right-hand limits at x = 5 and…

A: Now, Left hand limit at x = 5 is, And right hand limit is,

Q: Let f be a function with domain the set of real numbers R such that xf(x) +2 = f(x) + Vx2 + 3, for…

A: We can find the formula of f by just rearranging the terms and separating f(x) from the given…

Q: Show that f is a discontinuous function for all x, where f (x) is defined as follows: Show that f 2…

A: Here given function as fx=1for x rational-1for x irrational Also f2x=fx·fx

Q: If f is C2 on an interval prove that f(x + h) – 2f(x) + f(x – h) lim h-0 = f" (x). h2

A: We can prove this with the help of Lagrange Mean value Theorem as follows.

Q: Suppose f and g are functions with domain R. If both f and g are even but ƒ +g is odd, then prove…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the constant a such that the function is continuous on the entire real line. x² - a² if x * a…

Q: Let f(x) be the function defined by f(x) = x3 -1 for x < −1, f(x) = ax + b for −1 ≤ x ≤ 4, and f(x)…

A: Given f(x) x3-1 for x<-1 ax +b for -1≤x <4 x2+2 for x≥4

Question

Let f be a function defined on an interval I. All we know about f is that there is a constant K> 0 so that
| f (a) - f (b) | ≤ K | a - b | ^ 2 ,for all a, b ∈ I.
Show that f must be constant on I.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Fourier Transformation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.