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Let (X t) and (Y ,4) be two topological spaces. Where X={a , b , c} and T={X, 0 , {a} , {a b}} . Y={x ,Y , 2} and A={ Y , 0 , {x} , {X , y}} Let f(X ,t) - (Y , 4) b...

Question

Let (X t) and (Y ,4) be two topological spaces. Where X={a , b , c} and T={X, 0 , {a} , {a b}} . Y={x ,Y , 2} and A={ Y , 0 , {x} , {X , y}} Let f(X ,t) - (Y , 4) be a function defined by: f(a) =X f(b)=y _ f(c) =2 Then f is continuous at a

Let (X t) and (Y ,4) be two topological spaces. Where X={a , b , c} and T={X, 0 , {a} , {a b}} . Y={x ,Y , 2} and A={ Y , 0 , {x} , {X , y}} Let f(X ,t) - (Y , 4) be a function defined by: f(a) =X f(b)=y _ f(c) =2 Then f is continuous at a



Answers

Let $$f(x, y)=\left\{\begin{array}{ll} \frac{\sin \left(x^{2}+y^{2}-1\right)}{x^{2}+y^{2}-1} & \text { if } x^{2}+y^{2} \neq 1 \\ b & \text { if } x^{2}+y^{2}=1 \end{array}\right.$$ Find the value of $b$ for which $f$ is continuous at all points in $\mathbb{R}^{2}$.

There is a function off F by off to bury will as one minus goes X by a plus two x way a bone X ray. We have to find out the value off a very which the function is a continuous at all point off or Esquire. So simply we have to find out the limit of a function at point A so f x by speaker toe limit off the function off X by such as is equal to one minus goes X by upon x way bless two x by upon X y simply we have make the addition off the function. So we putting the value off hair off limit as zero vigor a value off zero Fill us to where we know that a limit off extending to zero for minus cause X upon X is always zero. This is a standard from hands. The value off a comes at two and eight dysfunction. It will be continous toe. All points off are is square this morning to Xing bite. So this is the answer off the question

In this problem, we want to construct a function F of X. Y. That is continuous everywhere on the xy plane, Except on the line, x equals two. So, A function that's continuous everywhere on the plane, except that points on this line where x equals two. Well, a continuous function is easy enough to create. We could do X times Y. Uh This would be continuous everywhere. Um including uh you know, any points where the X coordinate is too. So if we want uh to make a function that's going to be Discontinuous when x equals to start thinking about not being able to divide by zero. So let's try to create a function uh where the denominator becomes zero only when X is too well, we want a denominator and we want the denominator to be zero when X is too Well, if we do X -2 in the denominator, X -2 will be zero Whenever X is too. And this function will be discontinuous at points where the denominator is zero, specifically anytime access to the denominator is going to be zero and you're function is not going to be continuous. So for any points on the line, X equals two. Of course the X coordinate, four points on this blue line have an X coordinate up to. So for every point on this line X is equal to to, the denominator is zero, you can't divide by zero and the function will be discontinuous. It'll be continuous everywhere else because everywhere else X is not too. And so the denominator is not zero. So this is our function here is a function continuous everywhere, Except for points on the line whose equation is x equals two.

So because every skin genius on duh f not me coach is zero. It means that found into over, I would you be next in here. That's will be no fond see system that FFC equal zero. And that's why ah, because Emma's continuous in means that doesn't implies the effort X will be credited, ends off all angst in I need to be. It will be f x will be smaller than zero for all banks in I and B, they can look at this fall cushion in terms of the craft. Yeah, so we have every never e could do zero and I was continuous. So let's consider the Jew the in development and it should be here I and B and every never got the access s so it will be either floating above here. I didn't below this. I am being here, So nothing. Why, It would be either two cases here

Question of the 19 we have to sketch in party. We have to sketch the various possible graphs off the function. So the first part is pretty much constant. That's extra square minus three X plus three. So that is actually a parabola which is opening up words because coefficient of X square is positive and its vortex is given by a negative be over a B s and the coefficient off X which is minus three over a which is the coefficient of excess square, which is one. So the street Onda. If we consider this wire, then wired X equals three is nine minus nine plus three, which is three. So it means that the vortexes are 33 So if you're going to sketch the skull, these are the coordinate access. This is three and three. So if it is going toe, open up for certain will open something like this on. This is the Vertex are 33 But we know that this is not defined at X equal toe too. So we'll place ah hole at X equal to two thistles. X equal toe on will define the covert executed too. Like this. Okay and in fire. This won't really intersect the X the y axis as well. Actually, it will rather intersect the Y axis where X zeros affects zero than why would be why would actually amount us three. There's a correction in the, uh in the vortex because vortexes minus b over to a So this will be three over to. So if you place three over to here, that will be, uh, nine or four minus nine or two plus three, which is nothing but the common. Let's shift this a little bit down. We have to redraw this at least some part. Uh, and this welcome. Motors scared us up. We have the common. This will be negative. 9/4 plus three, which is nothing but 3/4. So the world Texas 3/2 and 3/4. So let's clear it up on draw this once again. So we have three or four here. So if this is the garden and access on 3/2 and 3/4 is the vortex that we come over here on, the graph will go something like this because that's a parabola. I will say that this point is three over to on. This is three or four and it actually going to do we have a whole So let's say this is actually going to do so. We have a whole over here on this corresponds toe execute toe. All right, now at X equal to we have a now it can be anything. So let's say can for the multiple graphs that can be over here it can be over here. Can be, uh these two are actually with the very same very same acts. Since this is equal toe y cual toe a so the value off viable. Keep on changing that as it can be on this a straight line there. They can lie anywhere so it can be over here. It can be over here. It can be over here. It may come in the hole. It can be here. It can be here and so on. So these are the multiple possible graphs off effects on in part B. They're asking the value of a so that it is continuous are too so clearly if a comes in this hole and it filters than the graph will actually become continuous function. So We'll say that the value of a must be world to graphically we have found out so that a phase tooth and the complete peace prize function will actually be continues. So the value off is the X coordinate off. This is too. While the value off they will be the value off dysfunction at X equals two. So the functional value of X equal to is we place executed to here. So we have f two as that will be two square minus three times two plus three. So that is four minus six plus three, which is nothing but one. So the value off is actually one in this case because the X coordinate at that point is to while the y coordinate Azzan the value off. This is one which should be the value off as well. So this is the answer. Wow.


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