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Q: Ex: Let (X, d) be a metric space and E, andE2 EX Show that whether each one of the following…

A: According to guidelines I solve first 3 subparts so kindly request you to repost the remaining…

Q: Suppose a and c are real numbers, c > 0 and f is defined on -1,1] by fxª sin(]x|=) to if x # 0) if x…

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Q: (7 ifx # y to if x y Prove d(x, y) = is a metric on R (real numbers). (9 points) V.

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Q: 5. Let d be a metric on X. Determine all constants k such that (i) kd, (ii) d+ k is a metric on X.

A: Given that d be a metric on X. Then d:X×X→[0,∞) satisfies the following conditions for x,y,z∈X, (a)…

Q: Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and…

A: we have  f=u+iv and g=p+iq  are analytic ⇒f and g satisfy the Cauchy Riemann equationux=vy and…

Q: 16.D. Let f and g be continuous functions on Rº to R and let h, k be defined for x in Rº by h(x) =…

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Q: 4. i) Let X = B [a, b] and let d be the sup metric on X. If n → x in (X,d) and an is continuous at…

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Q: Q.4// prove that: cothx = In, |x +1 -1 х —1

A: let y=coth-1(x) x=cothy We know that cothy=x=ey+e-yey-e-y xey-xe-y=ey+e-y Multiply ey on both sides…

Q: xMyn x2+y2 (х, у) 7 (0, 0) f(x, у) %D (х, у) %3D (0, 0)

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Q: Suppose Show that p: X²> t0,00) defined by d is a metric n X, 2. 3d (x,y) 2+3d (xy) metric an p…

A: Given that d is a metric on X. We need to prove that ρ is a metric.

Q: Drag the postulate or theorem that proves the trieng AZWX AY XW a. ATSU AXVW # SSS E SAS

A: which criteria follow to congruent triangle

Q: BLet f sohisfies the following data 3. 2. 2. Use Muller's mc th od to fnd a, st

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Q: 2.2.15. Let X be a metric space, and suppose that xn → I and yn → K. Prove that d(xn, Yn) – + y in…

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Q: Q2: In Euclidean metric space (R², 1. 1), is the set A = {(x, y): xy < 1} compact? giving the…

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Q: QNo 1 :- Suppose that fn f (un (umipormly) Set A s R. on (Peintwise ) A S R. cm

A: (i) Given that  fn→f (uniformly) and gn→g(uniformly) on set A⊆R. To show that fngn→fg is pointwise…

Q: The function u, takes on the following values : 1 u, = 4 1 + 4 1 1 1 Prove that lim u, .....u. 10…

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Q: (b) Using Identity principle, If f is analytic on the unit disk D(0,1) and f ÷) = 0 for all positive…

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Q: Check the validity of Rolle’s theorem for the functionf(x)=√1−x2on the segment [−1,1].

A: Under square root we can not have negative number So the function f(x) exists in the interval…

Q: 2. Do as indicated. if r m, r+1 a. Find all values of m such that g(r)= is continuous everywhere.…

A: Given function is: gx=x+1if x≤mx2if x&gt;m To find all values of m such that given function is…

Q: 4. Let (x, d) be a metric space and ACX be nonempty set. Define f:X → IR, fix)= inf dex,a). aE A…

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Q: Show using Definition 4.3.1(Continuity) that if c is an isolated point ofA ⊆ R, then f : A → R is…

A: If c∈A⊆ R is an isolated point of A ⊆ R. To prove  f : A → R is continuous at c. Let ∈&gt;0 , Since…

Q: 3. i) Let (X, d) be a metric space. If æn → x in (X, d) and y e X, prove that lim d(xn, y) = d (x,…

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Q: 18. Find numbers a and b so that r+ 4, ux + b, 3r 8, x> 3 1<r 3 is continuous everywhere.

A: we need to find a=?                          b=?

Q: Q1: In Euclidean metric space (R, I. I), prove that (1-2)→1 as n →∞o in R. cm² ID : (1) 13 compact?…

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Q: 8. PROVE: Suppose that SCR is connected, and that f : R → R is continuous. Then f(S) is connected.…

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Q: Q2: (a) In Euclidean metric space (R, 1. 1),if A = {y € R: y = 2 cos (2x); x € R} then fined the…

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Q: Bölom 2 1) Let is be defined E= {p€ Q:2 <p^< 3}. as Show that the se t E is closed and bounded in Q.…

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Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

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Q: a. Show that T: 1² → 1² defined by Tx = y = (n), n, = r, is compact where x = (5,) and where0 <r< 1.

A: Show that T: l2→l2 defined by Tx=y=ηj, ηj=rjξj is compact where x=ξj and 0&lt;r&lt;1. Definition:…

Q: Use Hölder inequality to prove l|u|la < llu||||;- for u E L" (N) 1 1-2 Where p <q <r and =

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Q: Find all volves of o,b ond c such that the func Ef×メ goo)= てメー1) 1チxg? 21 if x= 1 is continuous at…

A: Given, gx=ax3+bx2+cx+1x-12,  if x≠12,                               if x=1

Q: Suppose u is a finite Borel measure on R, which is singular to Lebesgue measure. Show that except…

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Q: - Pcosø prove that dw xy is Cont s Lony) - the fun ction (-1, 2) = x*y+z and x = PSinf cse y=pSi- -…

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Q: Prove the theorem stating that a E R, if x belongs to the neighborhood V¿(a) for every ɛ > 0, then x…

A: To prove the theorem stating that a∈ℝ, if x belongs to the neighborhood Vεa for every ε&gt;0, then…

Q: 9 on X= IR, k>o. Define metrics ond q P is not in the form kq for

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Q: 3. Let f, g : D →R be continuous at c € D. Prove that fg is continuous at c.

A: Since you have asked multiple questions, we have answered the first question. If you want the second…

Q: prove tらット IF Sequeree divergent to is THEN is boumdad below. the thif every conveng ent Sequenca…

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Q: S={xER:x2 - 4<0} with the usual metric is A. Compact. B. connected. C. Not connected D. Sequentially…

A: Given set  S= {x€R : x2 -4&lt;0}

Q: 2.9.11 Prove that every function f: Z→ F is continuous, where the metric on Z is the restriction of…

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Q: Let (X, d) be a metric space. Suppose that (r,) is a sequence of points in X which converges to a…

A: Given that X,d be a metric space. Suppose that xn is a sequence of points in X which converges to a…

Q: 2. (i). Let g : R→R be continuous at c and let C ={r€R:f(x)=0}. If c, EC and lim c, =c. %3D Prove…

A: PART A Let g: ℝ→ℝ be continuous at c. Let C={ x∈ℝ : f(x)=0}  Let cn∈C and  lim cn=c  As,  cn∈ C…

Q: 6. If af/ðæ is continuous on the rectangle R = {(t, x) : 0 0 such that |f(t, x1) – f(t, x2)| < K\x1…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Defn `x;→-∞ "xj diverges to minas infinity" 3Ng such that VjNg *; O Hww7 Let xj = - 8j prove Xj o…

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Q: 11. Verify Rolle's theorem for: c) f(x) = In () on [1,3] %3D

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