Question: (a) Show that the absolute value function F(x) = |x| is continuous everywhere. (b) Prove that if f is a continuous function on an interval,

(a) Show that the absolute value function F(x) = |x| is continuous everywhere.
(b) Prove that if f is a continuous function on an interval, then so is | f |.
(c) Is the converse of the statement in part (b) also true? In other words, if | f | is continuous, does it follow that f is continuous? If so, prove it. If not, find a counterexample.

Step by Step Solution

★★★★★

3.46 Rating (162 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a lim Fx 0 and lim Fx 0 so lim Fx 0 which is F0 and he... 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)

M-C-L-D (128).docx

120 KBs Word File

Students Have Also Explored These Related Calculus Questions!

Prove that, if f'(x) exists and is continuous on an interval l and if f'(x) 0 at all interior points of l, then either f is increasing throughout l or decreasing throughout l.

If f and g are increasing on an interval l, then f + g is increasing on l.

The absolute value function f(x) = |x|. The following functions all fail to be differentiable at x = 0. In each case, graph the function, see what happens if you try to compute the derivative as the...

a: f ( - 4 ) = B: f (0 ) = L: f ( 2 ) = 3 . Jududadded give the slope and the y- intercept of the live with the 4= 3x +4 given equation ! Graph the linear equation Use transformations of absolute...

Consider this absolute value function f x 1 8 How can function fbe written as a piecewise function O A O B O C O D f x f x f x f x 8 8 8 8 I 8 8 2 8 I 8 I 8 8 8 8 8 8 8

Please show work 10. Consider the function f (x) = x2-9 (x-3) (2-7) (a) Domain of f (x) in interval notation. (b) Vertical asymptotes of f(x) at_(write: DNE if none) (c) Horizontal asymptotes of f(x)...

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...

Hello, please answer all questions (1-4) seen below and ensure that you answer everything correctly. Please do not attempt these problems if you do not understand/can't answer them properly. These...

What is the motivation for violating the generally accepted accounting principle of matching revenues with expenses? What is the result of committing this fraud?

Consider you are working as a Senior Manager of a small-medium sized insurance private limited organisation, which deals with its customers all over the world to sell its motor, home, life, savings...

Why has the United States brewing industry become more concentrated over the last two decades?

Hassle-Free Web is bidding to provide Web-page hosting services for Hotel Lisbon. Hotel Lisbon pays its current provider $10,000 per year for hosting its Web page and handling transactions on it,...

Two surge tanks are used to dampen pressure fluctuations caused by erratic operations of a large air compressor. (See Figure) (a) If the discharge pressure of the compressor is Pd(t) and the...

The completed financial statement columns of the worksheet for Woods Company are shown below. Instructions(a) Prepare an income statement, owner??s equity statement, and a classified balance sheet.S....

In Table 1 , what was the second best resolution technique? What was the worst resolution technique? What do you conclude from this? Effective Resolution Ineffective Resolution (N = 53) (N = 53)...

Show that l, and 12 both commute with the Hamiltonian for a hydrogen atom. What is the significance of this result?

Larger animals use proportionately less energy than smaller animals; that is, it takes less energy per kg to power an elephant than to power a human. A 5000 kg African elephant requires about 70,000...

Sketch a direction field for the differential equation. Then use it to sketch three solution curves. y' = 1/2y

(a) dy / dx = xy2 (b) xy2 y' = x + 1 (c) (y + sin y)y' = x + x3

Find the solution of the differential equation that satisfies the given initial condition. (a) dy/dx = x/y' y(0) = -3 (b) du/dt = 2t + sec2t/2u, u(0) = -5 (c) x 1n x = y(1 + 3 + y2)y', y(1) = -1

In 5-card poker, find the probability of being dealt the following hand. Refer to the table. Note that a standard deck of playing Number of cards has 52 cards-4 suits (clubs, diamonds, hearts,...

Can someone help? For many years businesses have struggled with the rising cost of health care. But recently, the Increases have slowed due to less Ination in health care prices and employees paying...

Deployment Specialists pays a current (annual) dividend of $1 and is expected to grow at 18% for two years and then at 4% thereafter. If the required return for Deployment Specialists is 7.5%, what...