Question: A function f E R is called Lipschitz (or more precisely M-Lipschitz) if there exists an M >0 such that for all x, y

A function f E R is called Lipschitz (or more precisely M-Lipschitz) if there exists an M >0 such that for all x, y E, f(x)-f(y)| Mx - y. 1. Show that any Lipschitz function is uniformly continuous. 2. Show that if f: (a, b) R is a differentiable function such that f'(t)| M for all t (a, b), then f is M-Lipschitz.

Step by Step Solution

★★★★★

3.47 Rating (183 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

ANSWER To show that any Lipschitz ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

A function is called absolutely continuous on an interval I if for any > 0 there exists a > 0 such that for any pair-wise disjoint subintervals (xk, yk) = k = 1, 2, . . . , n, of I such that (xk -...

If lim tan(x-2)(x + (k- 2)x- 2k) ATIC 73 lfa,Bare the roots of the equation, the A+2 Jr. is equal to -4x +4 =5, then k 1-cos(ar +hx +e) (x-ay I) a (a-B) 1) 0 79. If (x) 2) 1 lim 3) 2 4) 3 2) 4a' (u-...

KYC's stock price can go up by 15 percent every year, or down by 10 percent. Both outcomes are equally likely. The risk free rate is 5 percent, and the current stock price of KYC is 100. (a) Price a...

answer the question clearly You are building a flight-control system for which a convincing safety case must be made. Would you assign the tasks of safety requirements engineering, test case...

I am looking for some help on finding the answers for these. Thank you! M2 Group Assignment Instructions Complete the Assignment, name it as GroupXX_Assign2.xls (where XX is your Group Name), and...

In Loewenstein and Thaler, the authors discuss a change in a West Virginia law that substantially reduced the number of high school dropouts. Briefly discuss this change, explain why it is surprising...

hello I have an exercise I am asked to summarize a text by Alain Touraine (globalization: realities, ideologies and decline.) that I have attached below I have no idea how to do the summary is what...

polynomial approximation. Calculate the error of each approximation. 5 Exercise. Find the fourth-order Taylor polynomial approximation for the function, g(x) = 2x cos(x), centered at x = T. (Use the...

Your discussion should include a brief summary (including the big ideas or major points of the article) along with some synthesis of the ideas and some of your own thoughts. You should draw any...

American Shogun GETTING STARTED It was in late May 2002 when Jan Vesely, sales manager for Southeast Asia and the Pacific region at International Instruments, Inc., received a call from RisingSun,...

b6-2 Forward exchange contract designated as a fair value hedge of a foreign-currency-denominated firm commitment to sell inventory, strengthening $US Our U.S.-based company enters into a firm...

A lab technician wants to measure the terminal voltage of a battery connected to a load resistor R-2 Q. The battery has r 0.50 internal resistance and emf e-5V. The lab technician being amateur...

Julia mark and Peter own a rental property as tenants in common. Julia owns a 40% share, mark owns a 35% share and Peter owns the remaining 25%. the property generates $2000 in rental income for the...

What is the probability that the first DVD taken from Roger's pack is defective? Unfortunately for Roger, the first DVD he chooses is defective! What is the probability that the DVD came from the...

1. Read the short story below. Crunchy TacoPotato. Paneer (cheese) and Potato Burritos.1 These names may sound strange unless you're a customer at a Taco Bell restaurant in India. Yum Brands Inc....

LLL Avionics Ltd. has contacted your accounting firm to inquire about the cost of an external audit. The company's president explained that he feels that the previous auditor charged too much and...

what is the best decision if you want to take the conservative approach? (also called maximin strategy) Trade, with a 1000 payoff Invest, with a 100000 payoff Trade, with a 0 payoff Invest, with a...

In the circuit shown in Figure 4, a battery supplies a constant voltage of 40 V, the inductance is 2 H, the resistance is 10, and l(0) = 0. (a) Find l(t). (b) Find the current after 0.1s.

Find all x R that satisfy the following inequalities. (a) |x - 1| > |x + 1|; (b) |x| + |x + 1|

Show that if (fn), (gn) converge uniformly on the set A to f, g, respectively, then (fn + gn) converges uniformly on A to f + g.

Show that a nonempty finite set S R contains its supremum. [Use Mathematical Induction and the preceding exercise.]

Scientist Robert Boyle examined the relationship between the volume in which a gas is contained and the pressure in its container. He used a cylindrical container with a moveable top that could be...

The following table shows stopping distances in feet for a car tested 3 times at each of 5 speeds. We hope to create a model that predicts Stopping Distance from the Speed of the car. a) Explain why...

At a meeting of the International Astronomical Union (IAU) in Prague in 2006, Pluto was determined not to be a planet, but rather the largest member of the Kuiper belt of icy objects. Let's examine...