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Let fu)ex'Tnen1s not continuous on 1-3, , |Tnen Is unbounded On [-} } |Then has no absolute extfemurO (-3.3 |Then has an absolute naxinun and absolute ALnaua O...

Question

Let fu)ex'Tnen1s not continuous on 1-3, , |Tnen Is unbounded On [-} } |Then has no absolute extfemurO (-3.3 |Then has an absolute naxinun and absolute ALnaua On [~}.} ]

Let fu)ex' Tnen 1s not continuous on 1-3, , | Tnen Is unbounded On [-} } | Then has no absolute extfemurO (-3.3 | Then has an absolute naxinun and absolute ALnaua On [~}.} ]



Answers

Continuity of the absolute value function Prove that the absolute value function $|x|$ is continuous for all values of $x .$ (Hint: Using the definition of the absolute value function, compute $\left.\lim _{x \rightarrow 0^{-}}|x| \text { and } \lim _{x \rightarrow 0^{+}}|x| .\right)$

Okay, so definition, absolutely. You function. I'm gonna treat it like a piece of ice function and just say absolutely I function. I could rewrite as after vax equals X, where X is greater than or equal to zero is equal to negative X, where X is less than zero. Okay, And then so to prove that its continuous then I need to say that, um, it's continuous for all values. Well, no. But you need to plug in, uh, just any value. So because they're polynomial, there are always gonna be equals. Has had to check the endpoints. Since f of zero equals zero. We're good there now, the limit as X approaches zero from the right breath of X. It's gonna equals zero. Because it's a polynomial and limit as X approaches here from the left, I will also equal zero because it's a polynomial. We can just plug it in. So because these limits are equal to each other Ah, and there equal to zero and f of a f of zero is zero. Then this is continuous for all values, because otherwise it behaves like those pollen. Emile's. Thank you very much.

With the given problem, we want to prove that the absolute value function is continuous for all values of X. And we're going to use the definition of the absolute value function to compute the limit as X approaches zero from the left and zero from the right. So really if we look at the absolute value function, we noticed that this is all clearly continuous. This is all clearly continuous because it's um we know that the absolute value function is just equal to X when x is greater than zero and negative acts when access lessons. Um So because of this, what we're going to have is the fact that our function um is more interesting right here and we want to know if it's continuous at this point, but we noticed that the limit as X approaches zero from the left and the right is equal to zero. And that's the same thing as the value of the function F of zero equals zero. So because of this, we see it as continuous for all X.

And question number 72.8. We can right ffx equal to X X is greater than or equal zoo or X equal to X f. Excuse us than you. So the function is a polynomial on the domain native infinity to zoo unions. Zero and thank you. Hence, by section 2.50 number seven, we conclude that F is continuous for all. It's not equal. Zero at X equals zero. We can clean the left and right hand limits. So let it. It's a big I think, safer to left hand zero equal to. Then it takes equal to zero. And then it's a few weeks exposed to right hand equal to men lifted eggs for equal to also zero. Since these are equal, we have there with a phobics where X goes to zero equal to zero. Finally, if it was you, his equals you. And since it's equal to the above limit, we conclude that if it's continuous, ah X equals zero. Hints is continuous on a tip. Infinity to infinity, moving to the point B. If two functions are continuous, then their composition is also continues. We can write absolute for equal to Capital Circle as where ethics equal to absolute value of X. In part, we prove that F of X is continuous and it's given that F is continued two, therefore composition that took two functions. It's also continuous talking to the point c a lot f X equal to one F X is greater than equal zero and negative one if X is less than zero and then a physical to one everywhere and hence continues. But if isn't continuous right zero and that's the end. Thank you.

That's no. Let's now prove that the absolute my liver function. So Michael backs equal absolute value of eggs. That's proof that it's continuous everywhere. Um so let's see. But we know that after backs is biz wise function. The absolute value can be were written as thanks the absolute value. Max is just equal eggs. If X is greater, equals zero Is Ikhwan minor sex if access best that zero so buff of these two functions ex linear functions X and minor sex. So they are continues everywhere. The only point of concern might be X equals zero, but it's clear that limits when x Porteous zero off after Max. What it's zero either from dried or from the left, is just zero or the time, and this is equal to divide you. Of the function at X equals zero is equal toe F zero absolute value up zero so that by using the definition of continuity, you can easily prove that the absolute value function, even if it's a piece wise function. But its anyway continues everywhere, zero included. No, let's try to prove in a more general away that if we know that general function after backs is continuous on a given in terminal done. We can also side that the absolute value off after backs is a continuous function. What? I think it's quite straightforward because a sui so here at the beginning, I mean we can always rewrite the absolute body function as a piece weighs function. So in some way the absolute value function. It's a composition. Off to function is a composite function where we do combine the hey function and the absolute body function. So it's does the composition of two functions end? They're just buff, continuous function, the absolute value function. We just show it continues function. And if we already know that after backs is continuous, well, we just have to remember that if f n g r two continues function than the composition off the two functions F and G, which meant off geo backs, it's still a continuous function. So in this case, uh huh, thinking about the absolute value on any function after bags. What if we call the absolute value function? Let's say we call it, uh, capitalized. Andi, we have just our function I fullbacks than in this case we can say that I've off capital. Life off smaller fullbacks is just a continuous function because it's a composite function and it's made of two continues function small off of axe, which we know is continues on absolute binding function, which, with does prove, is a continuous function. So just the composition up to continuous function. Now let's see, uh, if it's, we can also sigh What? That's clear a little bit if we can say it or so you know, they exactly the opposite. So can we say that the converse of this statement is also true? I mean, if we know that it is the absolute who Valley Off after backs is continuous, can we say dead have for bags is a continuous function. Question mark. What? Let's Maybe these this way is looking at an example and actually is a counter example, because if we just proved that there is an example where this is not true, then for sure, this this statement is not true. So let's look, for example, at the function. Careful banks, equal bees wise function. That's I for black sequin, for example, to if X is, we talked for a zero or and say minus two, If exist. Last No. Zero. What? Let's see. This is after backs. Let's think about the absolute value off after backs with the absolute value of F or backs you can. Ecstasy is always equal to, so it's for sure. Is constant is a continuous function. But what about Advil bags? What I feel backs is for sure, it's continues if we take the to pieces of the function what? They're just constant, so they are separately continues. But what about zero? Zero is It may be an issue, so let's check continue. It'd at zero. Well, we know that we can see that f of zero that by definition of the function is equal to. But what about the limit? With acts Porteous zero. Well, we have to distinguish limit from the left and from the right. So let's see limit from the left. But we're function after Amec's when X purchase. Zero. What? This is equal minus two and you may when X a purchase. Zero problem. The riot. My function after backs is equal to so there are not equal to each other. So the limits of the function does not exist when X approaches zero. So the function affects, isnot continues, is he's continues. So this is a good example to show that if model ists uh so absolute value of a function is continuous, then we can state that also affects is continous. This implication is not violent.


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