Question: Let f : R R be defined by setting f(x) := x if x is rational, and f(x) = 0 if x is irrational.

Let f : R → R be defined by setting f(x) := x if x is rational, and f(x) = 0 if x is irrational.
(a) Show that f has a limit at x = 0.
(b) Use a sequential argument to show that if c ≠ 0, then f does not have a limit at c.

Step by Step Solution

★★★★★

3.39 Rating (177 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Since fx 0 x we can take to show that fx 0 b If c 0 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)

829-C-L-D (462).docx

120 KBs Word File

Students Have Also Explored These Related Calculus Questions!

Let f : R R be defined by f(x) := 2x4 + x4 sin(1/x) for x 0 and f(0) := 0. Show that f has an absolute minimum at x 0, but that its derivative has both positive and negative values in every...

Let f : R R be defined by f(x) = x2 for x R. (a) Show that the inverse image f-1(I) of an open interval I := (a, b) is either an open interval, the union of two open intervals, or empty, depending...

Let f: R -> R be defined by f(x) = x2. For each of the following subsets B of R, find f-l(B). (a) B = {0, 1} (b) B = {-1, 0, 1} (c) B = [0, 1} (d) B = [0, 1) (e) B = [0, 4} (f) B = (0, 1} U (4, 9)

Let f : R R be defined by f ( x ) : = { x ^ 2 if x is rational, x if x is irrational. } Using the definition of continuity directly prove that f is continuous at x = 1 and discontinuous at x = .

a) Soit la fonctionf:RR dfinie par f(x)=x^48x^2 . Trouvez les trois nombres critiquesa,b,c defet donnez leurs valeurs exactes dans la premire colonne du tableau ci-dessous (en ordre croissant:a

3. (3 pts) Let f : R - R be defined by f(x) = x + (x), where R is the set of real numbers. (a) Find the range of f. (b) Is f onto? (c) Is f one-to-one

Let I := [0, 1] and let f : I R be defined by f (x) := x for x rational, and f(x) := 1 - x for x irrational. Show that f is injective on I and that f (f(x)) = x for all x I. (Hence f is its own...

Kindly give out correct solutions Consider an individual with preferences represented by the following utility function: U (x1, 12) = 0; 2, 2 The individual faces prices: p1 = 0.75; p2 = 1 And has...

Let f: R - R be the function given by f(x) = x+ 1 : x rational 1 5 - x : x irrational Using the e - 6-definition, show a) limx-2 f(x) = 3, b) For a * 2, limx-a f(x) does not exist

1. Let f: [1,4] R be defined by f(x) = 4x for 0x 2 and f(x) 2

Repeat problem 11-3-22 with argon replaced by neon, entering the chamber at the same temperature as helium (all other conditions remaining the same). Problem 11-3-22 Helium at 200 kPa, 20oC is heated...

Phillips Corporation, which manufactures shoes, hired a recent college graduate to work in its accounting department. On the first day of work, the accountant was assigned to total a batch of...

The IRS provides a web-based tool to help taxpayers determine whether they are eligible for the earned income tax credit. Locate this tool at the IRS website. Then apply the facts related to a...

Socially responsible marketing campaigns such as the Subaru one mentioned are becoming more visible over time. Identify two to three other campaigns that you have seen. What do they use as a...

Find all values of z such that (a) ez = 2; (b) ez = 1 +3i; (c) exp(2z 1) = 1.

Show that exp(iz) = exp(i) if and only if z = n (n = 0, 1, 2, . . .).

Show that (a) Log(-ei) = 1 - /2i; (b) Log (1 - i) = ln 2 - /4 i.

How can Rupert hedge his dollar exposure with a currency futures contract? What is the difference between a futures contract and a forward contract? Does the currency futures contract need to be...

Question 5: If X0CHF/USD=1.17 and then one year later X1CHF/USD=0.97 by how many percent did the USD appreciate/depreciate relative to the CHF ? 5.63%2.30%17.09%12.80% Question 6 2 pts Question 6: If...

Assume Stocks A and B have the following characteristics: Stock A B Expected Return 9.5% 15.5% Standard Deviation 33.5% 62.5% The covariance between the returns on the two stocks is .0015. 2. Suppose...