Exercise 1.6.37.(i) Show that every function f :R - R of bounded variation is bounded, and that t...

Question

Question

Exercise 1.6.37.(i) Show that every function f :R - R of bounded variation is bounded, and that the limits limoo f(x) and lim

Answer 1

Answer #1

et detined a fun ction FIR-IR hex you con see this, function is ere yo on IS () ConsideY any Continuou, function passing thr

Answer 2

Similar Homework Help Questions

20.1. Show that if f : [a, b R is of bounded variation, then it is integrable on [a, b]. ; (bi: i...

20.1. Show that if f : [a, b R is of bounded variation, then it is integrable on [a, b]. ; (bi: i)(DULO(ld rl. VZLilin.l.ld DILL. 20.1. Show that if f : [a, b R is of bounded variation, then it is integrable on [a, b]. ; (bi: i)(DULO(ld rl. VZLilin.l.ld DILL.
hint This exercise 5 to use the definition of Riemann integral F. Let f : [a,...

hint This exercise 5 to use the definition of Riemann integral F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
Question 2.1. . (i) Give an example of a function, f: R R, that is not...

Question 2.1. . (i) Give an example of a function, f: R R, that is not bounded. (ii) Give an example of a function, f: (1.2) + R, that is not bounded. (iii) Give an example of a function, f: R → R. and a set. S. so that f attains its maximum on S. (iv) Give an example of a function, f: R R , and a set, S, so that f does not attain its maximum on S....

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...
Analysis problem (b) Let f, q be defined on A to R and let c be...

Analysis problem (b) Let f, q be defined on A to R and let c be a cluster point of A i. Show that if both lim f and lim (f + g) exist, then lim g exists. c I>c ii. If lim f and lim fg exist, does it follow that lim g exists? -c (c) Suppose that f and g have limits in R as x -> o and that f(x) < g(x) for all x € (a,...
Given the function f(r.y) lim f(x, y) (ry)-+(0,0) a. Evaluate iii. Along the line y= r: i. Along the r-axis: iv. Alo...

Given the function f(r.y) lim f(x, y) (ry)-+(0,0) a. Evaluate iii. Along the line y= r: i. Along the r-axis: iv. Along y12 ii. Along the gy-axis: ,f(x, y) exist? If yes, find the limit. If no, explain why not. lim (a.)-(0,0) b. Does (0,0)? Why or why not? c. Is f continuous at d. The graphs below show the surface and contour plots of f (graphed using WolframAlpha). Explain how the graphs explain your answers to parts (a)-(c) above....

If {f {n} (x)} is a succession of continuous, bounded, defined functions in a compact and that converge punctually in said compact. Will it then be {f_ {n} (x)} succession of functions uniformly bound...

If {f {n} (x)} is a succession of continuous, bounded, defined functions in a compact and that converge punctually in said compact. Will it then be {f_ {n} (x)} succession of functions uniformly bounded? Demonstrate or give a counterexample.
1. Consider the function -F5 sin(r) for r f(x) =2 for 1< 3 2-25 for 3...

1. Consider the function -F5 sin(r) for r f(x) =2 for 1< 3 2-25 for 3 x2 -9x + 20 Evaluate the following limits You do not have to cite limit laws, but you must show how you arrived at your answer If a limit Does Not Exist, explain why. You should use oo or -oo where applicable Calculating the limit using L'Hopital's Rule will receive NO CREDIT. (a) lim f(x) r-+0 (b) lim f(x)= z-1 (e) lim f(z) (d)...
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E...

real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q. II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...

3. Let the function f be a real valued bounded continuous function on R. Prove that...

3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...