Question: For n 1, let gn and g be functions defined on E ( Rk into R, and recall that {gn} is said to converge

For n ≥ 1, let gn and g be functions defined on E ( Rk into R, and recall that {gn} is said to converge continuously to g on E, if for every x ( E, gn(xn) →g(x) whenever xn →x, as n→∞. Then show that if {gn} converges continuously to g on E, it follows that g is continuous on E.

Step by Step Solution

★★★★★

3.54 Rating (171 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

All limits are taken as n or subsequences thereof tend to Continuous convergence implies th... 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

Document Format (1 attachment)

742-M-S-P (6973).docx

120 KBs Word File

Students Have Also Explored These Related Statistics Questions!

Let g and h be real-valued functions defined on R for which «R g(x)dÎ¼ and «R h(x)dÎ¼ are finite, where Î¼ is a (Ï-finite) measure in R. Then...

For n ¥ 1, let f n be real-valued measurable functions defined on an open set E R containing 0, and suppose that | f n ( x )| £ M (

For n ( 1, let fn be real-valued measurable functions defined on (, and suppose that |fn (x)| ( M ( for x ( (. Then for any sequence {xn} with it holds ( 0 a.e. [(], where ( is the Lebesgue measure? ...

In this question assume that p and q are atomic formulae. (a) Compare and contrast path formulae and state formulae in temporal logic. [4 marks] (b) Describe and contrast the meanings of F(G p) and...

1. (For each of the following two functions f (n) and g(n), indicate whether f = O(g), or f = (g) or both (in which case f = (g)). (a) f (n) = n and g(n) = n/ 3 n. (b) f (n) = 100n + lg n and g(n) =...

201609 Math 122 Assignment 4 Due: Monday, November 14 or Tuesday, November 15, in class, before the lecture starts There are five questions (40 marks), and one bonus question (4 marks), on the two...

Could anyone explain time-complexity on 4.2 and 4.3? asteI OI SIOwer Suppose that some algorithm A has running time f (n) and that algorithm B has running time g(n), on all inputs of size n. Assume...

1. (60 pts True/False. Indicate whether the statement is true or false. If the statement is false, present an explanation or counterexample. (No explanation necessary for each true statement.) (2...

10 Green's functions for PDEs In this final chapter we will apply the idea of Green's functions to PDEs, enabling us to solve the wave equation, diffusion equation and Laplace equation in unbounded...

Math 636 Final - Quiz Component 1. The system of linear equations 2x2 x3 = 3 2x1 + x2 + 2x3 = 1 x1 + x2 + 3x3 = 2 (a) is consistent with a unique solution. (b) is consistent with infinitely many...

The managers of "Upscale' Promotions Ltd. need to decide regarding the venue to use for the launch of their existing new product line. They have developed a payoff table for three locations:...

Devise the synthetic steps to accomplish the following transformations. Use reagents you like. Use retrosynthesis steps to arrive to your answer. HO"

pina corp reported net sales of 6 5 2 0 0 0 , 7 2 2 2 0 0 , and 7 8 2 4 0 0 in the years 2 0 2 4 , 2 0 2 5 , and 2 0 2 6 respectively. if 2 0 2 4 is the base year, what is the percentage do 2 0 2 6...

In a functional organization: A. Power primarily lies with the project manager. B. Power primarily lies with the functional manager. C. Power is blended between functional and project managers. D....

Refer to the Century National Bank data. Is it reasonable that the distribution of checking account balances approximates a normal probability distribution? Determine the mean and the standard...

In item such as an increase in taxes, recall of elected officials, or an expansion of public services can be placed on the ballot if a required number of valid signatures are collected on the...

Geoff Brown is the manager for a small telemarketing firm and is evaluating the sales rate of experienced workers in order to set minimum standards for new hires. During the past few weeks, he has...

Hammond Supplies expects sales of 250,073 units per year with carrying costs of $4.87 per unit and ordering cost of $6.32 per order. Assuming the level of inventory is stable, what is the optimal...

A zero-coupon bond has a duration equal to its maturity. (True/False) Select one: O True False Duration is a useful number because it combines the effects of maturity, coupon and market rates to...

When we are dealing with factual matters, one way to counter bias from the anchoring heuristic is to Group of answer choices start with the anchor, multiply it by two, and then find the average of...