* Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x is continuous.

Q: (a) Assume that g(x, y) is a function of the form x²+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that…

A: (a) The given function is gx,y=x3yx2+y2  if x,y≠0,00             if x,y=0,0. To show: (i) gx,y≤xy…

Q: (a) Find the points (x ,y, z) in space, where the function f(x, y, z) = Vx² +y -1 Is continuous.

A: It is well known that the equation z=x2+y2 represents a circle with centre at the origin. The…

Q: If f is continuous function on la, bl, then zf(z)dz = = (z)dz. Select one: a. True b. False

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Q: Let f:R→R satisfy f(x+y) =f(x) +f(y), ∀x,y∈R. If limx→0f(x) exists, prove that f is continuous on R

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Q: (a) Assume that g(x, y) is a function of the form g(x, y) {} 0 if (x, y) = (0,0). (i) Show that…

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Q: is continuous at (0,0). 3. Show that X = [0, 1] with the discrete metric is bounded but not totally…

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Q: Let f function from (R,d ) to (R,d1) defined as d(x) = x , where d discrete metric and d1 .is the…

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Q: Isin , it (z, y) # (0, 0) 2. Determine if a function f : R → R defined by f (r, y) = 0, if (z, y) =…

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Q: Tim 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: At what points (x, y, z) in space are the functions in continuous? h(x, y, z) = ln (z - x2 - y2 - 1)

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Q: Let f : [0, 1] → R be a continuous function. Show that the function g : [0, 1] → R given by g(x) =…

A: The given function: is continuous. That is, assume f(x)=x, then x is Real number. Show that the…

Q: Let f function from (R,d1) to (R,d2) defined as d(x) = x, where d, is the usual metric and d2 is the…

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Q: Jerify Divergence Theorem for F = = (x + y') i – 2 xỷ + 2 yzk and the volu

A: Given: F⇀=(x+y2)i^-2xj^+2yzk^The plane, 2x+y+2z=6

Q: Let a function f(x,y)=(x^2+2xy+y^2)/(x^2+y^2 ), find the limit over f(x,y) along the path (a) x-axis…

A: (x,y) = (x2 + 2xy +y2)/ x2 + y2 = (x+y)2/ x2 + y2   .x-axis .y = 0 .f(x,0) = x2/x2 = 1   1 =…

Q: 2. Suppose that f (x, y), g(x, y) are two continuous functions on the xy plane. Is it true that So…

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Q: If f : a, b] → Ris differentiable at a point c E (a, b), then f is continuous at c. True False

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Q: Determine whether the statement is true or false. If C is a smooth curve in the xy-plane and f(x, y)…

A: We have to determine whether the statement is true or false.

Q: f(z) = Im z is continuous 0 صواب ihi O

A: True

Q: Let f:X → Y be function and g: X → fcX ×Y, g(x) = (x, f(x)). Prove that if g is a homeomorphism then…

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

Q: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by f(r, y)…

A: See the answer below

Q: \(d\) is a metric on \(X\). Show that \(\rho : X^2 \to [0,\infty)\) defined by \[ \rho(x,y) =…

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Q: Let f: X_nte, Y be X,Y are Tj spaces. show that: Xö locally compact iff Y is so . perfect function,…

A: Given: f:X→ontoY is a perfect function, where X,Y are T2-spaces. To show that X is locally compact…

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: Suppose that X P(X < 3Y). unif(-2, 2) and Y ~ unif(-1,2) are independent continuous RVs. Find P (|X…

A: Given: X ~ U (-2, 2) Y ~ U (-1, 2) X and Y are independent.

Q: Determine if the function √x² + y² +1 x² + y² f(x, y): 1 2 is continuous at (x, y) = (0,0). If not,…

A: To find whether the given function is continuous or not at(0,0).

Q: Let X be the set of all continuous functions defined on the interval [a,b]. Let d: X × X →R be a…

A: We need to prove the all three conditions for metric.

Q: กำหนดให้ f(x, y) = xy cos.x x² +3y²

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Q: Determine the set of points at which the function is continuous. if (x, y) # (0, 0) 5x? + y? f(x, y)…

A: We know that given function is continuous in its domain so we need to check only at point where…

Q: 2. Determine the points where f is continuous where x + Y if (x, y) # (0,0) f(x, y) = x2 + y2 if (x,…

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Q: å:RR→R be defined by dcx.y)=|xl+1yl, HxixR (R,d)is Than a metric space.

A: As per our guidelines we supposed to solve the very first question when the multiple questions are…

Q: I. Show that the function f defined by 1, (x, y) = (1, –1) %3D f(r, y) = { r2 + y (x, y) + (1, –1)…

A: Given function is  fx,y=1,             x,y=1,-1x2+yx+y,    x,y≠1,-1

Q: (b) Let X = {r € R: -1 <1<0or 0 <1< 1} with the usual metric.Then X is disconnected.

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Q: At what points (x,y.z) in space are the functions continuous? a. h(x,y,z) = In (2z - 4x – 5y3 – 1) 1…

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

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Q: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by 2.x2 f…

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

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: Consider the function f defined by f(r,y) = y %3D lim(z,y) (0,0) f(x, y) =

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Q: Show that the function f defined by 1, f(r, y) = { r2 + y (x, y) = (1, –1) (x, y) # (1, –1) is not…

A: We have to show that  f(x, y)  is not continuous at  (1, -1)

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

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Q: /Evaluate The are bounded by The Semiellipes y=25-x' oP The lines X = ±1,Y=-1

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Q: Find and describe the points in the space that the function f(x, y,z)=In(xyz) is continuous.

A: We have to find and describe the points in the space that the function f(x,y,z)=ln(xyz) is…

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y,z) = In (2z2 – x² – 3y³ – 4)…

A: we want to find the domain of given  function

Q: (a) Assume that g(x, y) is a function of the form 2+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that…

A: As per our guidelines we are supposed to answer only first question's first 3 subpart. In this case…

Q: Determine if the function f (x, y) = if (x, y) = (0,0) is continuous at (x, y) = (0,0). If not,…

A: Use limit to find wheather function is continuous or discontinuous.

Q: (b) Let X = {x e R: -1 < x < 0 or 0 < x < 1} with the usual metric.Then X is disconnected. %3D

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Q: flz) = Im z is continuous %3D

A: The objective is to check f(z) =Im Z is continuous Or not.

Q: (a) Assume that f(r, y) is a function given by ry if (x, y) # (0,0), f(r, y) = if (x, y) = (0,0).…

A: We  need to prove the continuity of the function at (0,0).

Q: Plet SsR andl di SxS- R be a by dcx,y) = |x, -y,l+|xz-Yzl for all x=CX,rx2),y=G,%]E! Then (s.d) is a…

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Q: Show that if u = x – y and v = x + y, then for any continuous f, -2sx * f(x – y, x + y)dydx = / |…

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