Let Y be any topological space and y: [0, 1] → Y be a continuous function so that r(0) = r(1) = xp. Define e: [0,1] → Y by the rule e) = x, for all 0 sjs1. Show that y- ezo *, Y.

Q: Define an inner product space for continuous functions on [0, 1] as (S, g) = [f(x)g(x)ax with the…

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Q: Show that the following functions are continuous on their naturaldomains.

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Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(T)<0.

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Q: Let A = { (x,y)e RXR; 0<max{}x-31. ly-21)<1}, then: %3D None of the choices OA is not compact…

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Q: Define h(0,0) in such a way that h is continuous at origin       h(u,v)=ln(nu2-u2v2+nv2/u2+v2

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Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + Y (x, y) # (1, –1) x +y is…

A: Here, the given function is: f(x,y)=1;   (x,y)=(1,-1)x2+yx+y; (x,y)≠(1,-1) lim(x,y)→(1,-1)

Q: a) F(x,y, z) : planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 1. 3xi + xyj + 2xzk and E is the…

A: Solution of (a) is in given below

Q: Let f be a continuous function with domain R. Compute the limit (z- t) f(t) lim dt.

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Q: In Exercises 23-28 explain why fis continuous. 26. f(x, y, z) = sin (xyz – 1) 27. f(x, y, z) = ln(e"…

A: The function is continuous if all partial derivatives exist.

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Q: Define h(0,0) in such a way that h is continuous at origin nu? – u?v² + nv² u? + v? h(u, v) = In

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Q: Let if (x, y) # (0, 0) f(x, y) = { if (x, y) = (0, 0). (a) Compute the limit as (x, y) → (0, 0) of f…

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Q: If u = x -3xcy, then show that there exists a function V(x, y) such that W = u + iv is analytic in a…

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A: The given statement is True.

Q: Let f : [0 , 1] → R be a continuous function on [0 , 1] such that : f(x)x" dx = 0, (n = 0, 1, 2,…

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Q: If fx and fy are each continuous in an open region R, is f(x, y) continuous in R? Explain.

A: If fx and fy are each continuous in an open region R, is f(x, y) continuous in R? Explain.

Q: 2. Show that the function g defined by VVT y² + √² 2, is not continuous at (0,0). (z,y) = (0,0)…

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Q: Show that the function f defined by 1, (x, y) = (1,–1) f(r, y) = { x² + y (x, y) # (1,–1) r+ y is…

A: In this question, the concept of limit is applied. Limit Limits are the points at which a function…

Q: Let f be a bounded function on [a,b] and suppose that (P)aEN is a family of partitions on [a,b]such…

A: Given: Let f be a bounded function on [a,b] and (Pn)n∈N is a family of partitions on [a,b] such that…

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A: we need to find those points where the denominator ( z^4+1) equals to 0

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Q: (a) Let Ty if +y +0, g(x, y) = 0 if r = y = 0. Prove that g(r, y) is continuous at (0,0).

A: To Prove: g(x,y) is continuous at (0,0)

Q: If f (z) is analytic and f' (z) is continuous at all points inside and on a simple closed curve C,…

A: Solution:

Q: Find an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) → 0…

A: To give an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) →…

Q: Suppose f(x, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: Based on given data,For Non empty open region R,fx, y is continuous,andfx, y is non -…

Q: Let Y be any topological space and y: [0,1] → Y be a continuous function so that y(0) = r(1) Show…

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Q: Define T: (C¹ [0, 1], || · ||…) → (℃ [0, 1], || ||) by Tf = f'. Show that T is not continuous. Is…

A: T:C'[0,1], .∞→C'[0,1], .∞ T(f)=f' Let f(x)=xn then T(x)=nxn-1 T(x)=nxn-1 as n→∞,   T(x)→∞ So T is…

Q: Assume g(x,y) is a function of the form x³y g(x,y) = √²x²y³² ( O 1g (x, y)| ≤ lally! then < Show…

A: Solution ; g(x,y)=x3yx2+y2,(x,y)≠(0,0)0,(x,y)=(0,0) Use parametrization to prove required result,…

Q: If f(z) = u + iv is entire and u(x, y) is bounded above, apply Liouville's Theorem to exp(f(z)) to…

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Q: Let f: X→Y, g:Y-Z be uniformly continuous. Then gof:X-Z is uniformly continuous.

A: The main objective is to proof that the function g∘f is uniformly continuous, i.e. for every ε&gt;0,…

Q: Let f: (0, 1] x R R be continuous, define g : R R by g(v) = f(x, u)dr, %3D and suppose is continuous…

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Q: Define f : [0, ∞) → [0, ∞) by f(x) = Vx. (a) (b) Prove that va +b 0. Prove that f is uniformly…

A: We have to prove that   √a+√b≤ √(a + b) for all a, b ≥ 0.

Q: Let f : AC X → Y be a continuous function between metric spaces. If A is open then so is f(A).

A: Continuous Function: Let (X, dX) and (Y, dY ) be metric spaces. A function f : X → Y is continuous…

Q: Let X and Y be jointly continuous with density fxy (x, y) = 2xy for 0 < 2y < x < 2. Then E (XY) is…

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Q: Assume that J is continuous on [0, 3 and differentiable on (0, 3) with f (z) > 6 for each z in (0,…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

A: Given function is fx,y=1,           x,y=1,−1x2+yx+y  ,x,y≠1,−1 . We have to show that the given…

Q: f(z) = Im z is continuous ylgn O ihi O

A: Use the property of continyity in C which is if real and imaginary part of functiom is continuous in…

Q: . Suppose f(a, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: According to question,f(x, y) is continuous,fx, y ≥ 0        non - negative,∫∫Rfx, y dA = 0

Q: Let x = s + t and y = s - t. Show that for any differentiable functionf (x, y),

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