4

3. Let f: X - Y be an injective function that is continuous. Let A be subset of X. Suppose that a is a limit point f A in X. Show that f(a) is a limit point of f(A)...

Question

3. Let f: X - Y be an injective function that is continuous. Let A be subset of X. Suppose that a is a limit point f A in X. Show that f(a) is a limit point of f(A) in Y.

3. Let f: X - Y be an injective function that is continuous. Let A be subset of X. Suppose that a is a limit point f A in X. Show that f(a) is a limit point of f(A) in Y.



Answers

Show that the function $ f $ given by $ f(x) = | x | $ is continuous on $ \mathbb{R}^n $. [ $ \textit{Hint:} $ Consider $ | x - a |^2 = (x - a) \cdot (x - a) $.]

In order to show that this is continuous. We have to show that it fits three conditions. 1st, the function has to exist at the the at all points. Now, there's only really one point in question here, and that's when it switches between uh X being greater than zero or X being less than or equal to zero. The function value at zero would come from the top functions and that's when X is less than or equal to zero. And if we fill in zero for X there, The function of zero. The second condition we have is that the limit as X approaches zero must exist. Now. In this case, we have to look at both sides, since there are two different functions That approach 0. 1 from the from the negative side, one from the positive side, The negative side of zero would be from the left. So we're going to fill in it into A X. And the positive. We're going to fill it into square root of X. You fill in zero to the first. You do get 00 times a. And the 2nd 1 You get Square 20, which is zero. So the limit is in fact zero. Third condition of continuity is that the function and the limit have to be the same values and both points. And that is true. Zero does equal zero, so it continues with zero and it was continuous at all. Other non zero point, since this is a continuous function And this is continuous with X is greater than zero. Mhm.

Here were given a function f x Geico gender one if for ex interational number ik O J minus one for X in the irrational number here and now we consider the new functional Kanis egy X Now do you fight as a f x Dudley square? And now let me consider at X acreage in any constancy and therefore have apps See GFC iko Joe Because that's the way we defined function f x I'll see square so it can be either one in a minus one square An echo to one on the country one that founded GFC We could you one on now The second case would be the limit on the function G off explain X goes to see here every coaches left on garaged emit and I said, it goes Judah one square and it could you one. And because job Gambia ico Doesn't he blast that g X echo Judah fx during the square is continua continuous because, say, here is any constant search continues for X in the real number

So I know that according to the laws punt, nudie, any basic functions and which the absolute value falls under our continuous. But we will be using three conditions to check that this function of X equals as the value of X is continuous for all values of X. So the first condition that will do well will be using at Sequel. Zero's a reference point. We need to make sure that this function is defined. X equals zero. So let's check that out real quick. Half of zero equals the absolute value of zero, which is simply zero. So that checks out. The second condition that we must check is the limit. As X approaches, zero from the left and the right now only exist but are equal to each other. So the limit as X approaches zero left hand side of the of acrobatics, which is the absolute value of X equals s. So I drew a graph on the right hand side just as a point of reference, and we see that a equals zero and now if we apply the same method for the right hand side, the limit as X approaches zero from the right hand side, uh, af of ax again. If we look at this little sketch Teo the right of my work, it also approaches zero. So both of these limits for both the left and right hand side exist and are equal to each other. So the second condition checks out. And lastly, we need to make sure that the limit as X approaches zero is equal to this defined value of half of zero, which, as we know, does. And just to make this clear, let's see limit as Ex purchase here of death, ADEX is indeed you're right here, which also equals of zero, which checks out as well.


Similar Solved Questions

4 answers
Pre-laboratory Asslgnment Must be turned in_ the beglnning of the lab:Titration curvesComplete the following acid base reactions. Write the net ionlc form; Determine if the solution made by mixing equal amounts moies of these acids and bases will be acidic, basic Or neutral. equations Explain Show relevant4.) NaOH(aq) HCI(aq)b.) NaOH(aq)HF(aq)NH;(aq) HCIaq)
Pre-laboratory Asslgnment Must be turned in_ the beglnning of the lab: Titration curves Complete the following acid base reactions. Write the net ionlc form; Determine if the solution made by mixing equal amounts moies of these acids and bases will be acidic, basic Or neutral. equations Explain Show...
5 answers
Penlom E 1 1 W 1 1 1 1 1 L 1 1 1 1 1 1l L tmjun 1 JH 1 V coteule 1 Jedll thet Ee 1 UEAEA 1 prutRa1 18 0 8 &
Penlom E 1 1 W 1 1 1 1 1 L 1 1 1 1 1 1l L tmjun 1 JH 1 V coteule 1 Jedll thet Ee 1 UEAEA 1 prutRa 1 1 8 0 8 &...
5 answers
2 attempts leftCheck my workClick in the answer box to activate the palette:Write the formula for the ionic compound formed from each cation and anion:ammonium and phosphateNH; PO 4
2 attempts left Check my work Click in the answer box to activate the palette: Write the formula for the ionic compound formed from each cation and anion: ammonium and phosphate NH; PO 4...
5 answers
Q2. [7] Consider the IVP: tan Y , 0 st <1, y(0) = 1, Find the value of h required by Euler's method to ensure that ly(1) - Wvl < 0.01
Q2. [7] Consider the IVP: tan Y , 0 st <1, y(0) = 1, Find the value of h required by Euler's method to ensure that ly(1) - Wvl < 0.01...
2 answers
Roblci Coucse Numecical Analysos ) The funckion fcx) _ i$ In(x) 3iven , Use #6e Com "posik Sim psens rule wil 4Ge m: 2 k functton On 4e intenval iakjrqk Cx, 6] ,
Roblci Coucse Numecical Analysos ) The funckion fcx) _ i$ In(x) 3iven , Use #6e Com "posik Sim psens rule wil 4Ge m: 2 k functton On 4e intenval iakjrqk Cx, 6] ,...
5 answers
Chapiet 03, Probinnt 00)1QlI Meemcomnonenirectorcomennent(a) nhatis the mantud7 (6) wha kthe mjeEanetn detton dItethapra drtoen d4nlemeplel(0} NumberrplamNumbaUnits[LLmNodt this auetLod= mSaLha
chapiet 03, Probinnt 00) 1QlI Meem comnoneni rector comennent (a) nhatis the mantud 7 (6) wha kthe mjeEanetn detton d Itethapra drtoen d4 nlem eplel (0} Number rplam Numba Units [LLm Nodt this auetLod= mSaLha...
5 answers
S7on that tne - following sanes diverges: Which condition of the Alternating Serios Test not satisfied? Z(-p+ 2k + 1Let a 2 0 represenl the magnitude of Lhe leris of the given series. Identlfy and describe Seloct the correct choice beiow and any answrer Dox your choice,and for any index thero are Kome values olk>whichand some values ofk > Nfo which -Ban incroating funclion for all k docreasing (unction fur all k
S7on that tne - following sanes diverges: Which condition of the Alternating Serios Test not satisfied? Z(-p+ 2k + 1 Let a 2 0 represenl the magnitude of Lhe leris of the given series. Identlfy and describe Seloct the correct choice beiow and any answrer Dox your choice, and for any index thero are ...
5 answers
Fricipi= 1-1.14pli0 H440 Fexm NVC,l,o
Frici pi= 1-1.14 pli 0 H 440 Fexm NV C,l,o...
5 answers
EdjeH paaNoot jood bujwwims 1-O€-Aq-Y-0z241 Pue 'Iood 041 Jo Jauuoj #UObuilie) 'Sen i31ui #-$paunseeWWdeDVocpaiiv900 ! 51 831835 Siarsi snoixaldQuicd MoedjoH paaNPiios 941 $0 #Wnioa 341 e1ewnsa aina JulodPIw 341 250'ajenbs 4JpaIauot 14bu Joddn 041 aquiod a dwes 04 a7e1=0'€44M uS uueubiy Osn{95452'95*501('} =& naoqe pue Ax Ojeuns 841 moia4 5241 141 Piios 841 Jo aunioa 04} B1EWp53Ibuepal buimolloj
edjeH paaN oot jood bujwwims 1-O€-Aq-Y-0z 241 Pue 'Iood 041 Jo Jauuoj #UO builie) 'Sen i31ui #-$ paunseeW WdeD Voc paiiv 900 ! 51 831835 Siarsi snoixald Quicd Mo edjoH paaN Piios 941 $0 #Wnioa 341 e1ewnsa aina JulodPIw 341 250 'ajenbs 4Jpa Iauot 14bu Joddn 041 aq uiod a dwes 04 ...
5 answers
0,51,52 (Bound](um)25Estimate Kd: OA 0.5 uM 08. 1.0 uM Oc 1.5 uM D: 2.0 HM OE 3.0 uM OF. 10 +M
0,5 1,5 2 (Bound](um) 25 Estimate Kd: OA 0.5 uM 08. 1.0 uM Oc 1.5 uM D: 2.0 HM OE 3.0 uM OF. 10 +M...
5 answers
In Problems, find the gradient of the given function at the indicated point.$$f(x, y)=x^{2}-4 y^{2} ;(2,4)$$
In Problems, find the gradient of the given function at the indicated point. $$ f(x, y)=x^{2}-4 y^{2} ;(2,4) $$...
5 answers
Problem 4. [20 points] Prove the following statement: Iff : R ~ Ris a function with f(x) > 0 for all x € R, then f (x) is strictly decreasing il and only il g(x) = If(x) is strictly increasing:
Problem 4. [20 points] Prove the following statement: Iff : R ~ Ris a function with f(x) > 0 for all x € R, then f (x) is strictly decreasing il and only il g(x) = If(x) is strictly increasing:...
5 answers
Examples: for {90, 100, 110} for discrete variable and [1. 5], "(0,1), 4] for continuous or [0, 1) or (5, variableHow many days in the week do you go to the gym? So the variable X is the number of days in the week that_ a student goes to the gym.
Examples: for {90, 100, 110} for discrete variable and [1. 5], "(0,1), 4] for continuous or [0, 1) or (5, variable How many days in the week do you go to the gym? So the variable X is the number of days in the week that_ a student goes to the gym....
5 answers
Question 18 ptsA linear constant coefficient differential equation is given by dyt) dylt) dx(t) +4- + 3y(t) +5c(t) dt2 When the forcing function (the input) is x(t) = e 2tu(t) , determine the total solution y(t) = yn (t) + Yp (t). The initial conditions are given bv y[0] =6 and dy(o) =5 Enter the value of the total solution at t= 0.412,i.e: y(0.412)- Enter your answer with three decimal places accuracy:
Question 1 8 pts A linear constant coefficient differential equation is given by dyt) dylt) dx(t) +4- + 3y(t) +5c(t) dt2 When the forcing function (the input) is x(t) = e 2tu(t) , determine the total solution y(t) = yn (t) + Yp (t). The initial conditions are given bv y[0] =6 and dy(o) =5 Enter the ...
1 answers
Solve each system of equations by graphing. See Example 1. $$ \left\{\begin{array}{l} x^{2}+4 y^{2}=4 \\ x=2 y^{2}-2 \end{array}\right. $$
Solve each system of equations by graphing. See Example 1. $$ \left\{\begin{array}{l} x^{2}+4 y^{2}=4 \\ x=2 y^{2}-2 \end{array}\right. $$...
5 answers
Graphing a Hyperbola, find thecenter, vertices, foci, and the equations of the asymptotesof the hyperbola. Use a graphing utility to graph thehyperbola and its asymptotes.$$25 x^{2}-4 y^{2}=100$$
Graphing a Hyperbola, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. $$ 25 x^{2}-4 y^{2}=100 $$...
5 answers
At-14.0 "C_ common temperature for houschold freezers, what the maximum mass of aspartame (C_-HJsN;Os) you cin edd to 00 kg of pure watcr and still have the solution freeze? Assume that aspartamc molecular solid and docs not ionizc when dissolves Wnicr Consult the table of VusMa9aspartame:
At-14.0 "C_ common temperature for houschold freezers, what the maximum mass of aspartame (C_-HJsN;Os) you cin edd to 00 kg of pure watcr and still have the solution freeze? Assume that aspartamc molecular solid and docs not ionizc when dissolves Wnicr Consult the table of Vus Ma9 aspartame:...
5 answers
While on a train moving east at a speed of (24.0 + A) m/srelative to the tracks, a passenger walks to the back of train at aspeed of 1.60 m/s relative to the train. Relative to the ground,how far east does the passenger travel in (32.0 + B) seconds?Calculate the answer in meters (m) and rounded to three significantfigures.A=0 B=20
While on a train moving east at a speed of (24.0 + A) m/s relative to the tracks, a passenger walks to the back of train at a speed of 1.60 m/s relative to the train. Relative to the ground, how far east does the passenger travel in (32.0 + B) seconds? Calculate the answer in meters (m) and rounded ...
5 answers
QuestionA force F = (4x + 3y)v acts on an object as it moves in the x direction from the origin to x=6m The work done on the object by the force isNot yet answeredMarked out of 1,00Select one: 72 JFlag " questionb. 78 J88 Jd. 66 JPrevious page
Question A force F = (4x + 3y)v acts on an object as it moves in the x direction from the origin to x=6m The work done on the object by the force is Not yet answered Marked out of 1,00 Select one: 72 J Flag " question b. 78 J 88 J d. 66 J Previous page...

-- 0.018953--