Question: Let be a function of bounded variation on [a, b] and be its total variation function. Prove that if is continuous at

Let ϕ be a function of bounded variation on [a, b] and Φ be its total variation function. Prove that if Φ is continuous at some point x0 ∈ (a, b), then ϕ is continuous at x0.

Step by Step Solution

★★★★★

3.34 Rating (151 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

By Theorem 556 x x 0 x x 0 If i... 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)

741-M-N-A-D-I (355).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

Suppose that f is continuous and of bounded variation on [-, ]. Prove that SNf f pointwise on (-, ) and uniformly on any [a, b] (-, ).

Let L R. A series k=0 ak is said to be Abel summable to L if and only if a) Let Sk = j=0 ak. Prove that provided any one of these series converges for all 0 b) Prove that if k=0 ak is Cesaro...

Suppose that and are of bounded variation on a closed interval [a, b). a) Prove that is of bounded variation on [a, b] for every a R. b) Prove that is of bounded variation on [a, b]. c) If there...

7. Let be a function of bounded variation on [a, b] and 1> be its total variation function. Prove that if 1> is continuous at some point Xo E (a, b), then is continuous at Xo. 00. This exercise is...

I am really desperate for someone to answer all 4 parts of this question (d, e, f, and g) (I have already posted this question twice and no one has answered all 4 parts sufficiently). This is due in...

( c ) Show that every infinite bounded subset of R has at least one limit point. d . Let f , g:R - > R and x _ ( 0 ) inR \ epsi lon - \ delta \ lim _ ( x - > x _ ( 0 ) ) f ( x ) = L and \ lim _ ( x -...

Math 325 Elementary Analysis Fall 2016 University of Nebraska-Lincoln Assignment #8 Hard deadline: Tuesday, December 6 at 10:30am Will be graded out of 60 points (20 points for each problem) 1...

(3) If [a, b] is a closed interval with -co 0 for each j = 0, . . . , n -1, and if f : [a, b] - R is a real function we write Aff = f (cj+1) - f (x;). We say that f is of bounded variation in a, b if...

Math 104A Homework #3 Instructor: Lihui Chai General Instructions: Please write your homework papers neatly. You need to turn in both full printouts of your codes and the appropriate runs you made....

Briefly describe ASCII and Unicode and draw attention to any relationship between them. [3 marks] (b) Briefly explain what a Reader is in the context of reading characters from data. [3 marks] A...

What is accounting and define its branches?

Use the data in cps91.dta for this question. These data are for married women. nwifeinc is the woman's nonwage income (i.e. other sources of income, which includes husband's earnings) measured in...

German physiologist Johann Blumenbach created the term "Caucasian" to describe members of the white race, basing the choice on _ _ _ _ _ .

A table from the Department of Labor with the real minimum wage in the United States can be found at: http://www.dol.gov/ILAB/media/reports/oiea/wagestudy/ FS-UnitedStates.htm. A better alternative,...

8. [IMPLICIT METHOD]. (a) Suppose that f, g : R3 - R are differentiable at a point (a, b, c), and f (a, b, c) is an extremum of f subject to the constraint g(x, y, z) = k, where k is a constant....

7. Let a, b, c, D, E be real numbers with c =I- O. (a) If DE > 0, find all extrema of ax + by + cz subject to the constraint z = Dx2 + Ey2. Prove that a maximum occurs when cD

6. Let V be an open set in Rn, a E V, and f : V ---+ R be C2 on V. If f (a) is a local minimum of f, prove that D(2) f (a) (h) 2: 0 for all h ERn.

The Framingham Company expects to grow at 4% indefinitely. Economists are currently asserting that investment opportunities in short term government securities (Treasury bills) are readily available...

Ironwood Bank is offering a 2 5 - year mortgage with an APR of 5 . 8 0 % based on monthly compounding. If you plan to borrow $ 1 6 6 , 0 0 0 , what will be your monthly payment? The loan payment is $

The following spreadsheet contains monthly returns for Cola Co . and Gas Co . for 2 0 2 2 : Using these data, estimate the average monthly return and the volatility for each stock. The average...