2. Let (X,r) and (Y.A) be two topological spaces. Let B be a base for A. Prove that a function f :(X,r)(Y,A) is continuous if and only if the inverse image under of every member of B is a r-open set.

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Q: Let X, Y be Banach spaces and U be an open subset of X. +1)-times continuously differentiable…

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-Df(x)+(y) vx, yER. Show that…

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Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

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Q: Let f: U C be a holomorphic function, where UCC be an open subset in the upper half-plane. Let V =…

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)3Df(x)+f) x, yER. Show that f(x)3Dcx…

A: Since you have posted a multiple questions. So,  we will solve only first question  for you. To get…

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

A: Given that, f:X→Y be function and g:X→f⊂X x Y defined byg(x) = (x,f(x))Given that, 'g' is…

Q: Let CR be the space of all continuous and bounded functions x(t) on R with norm ||x|| = supR |x(t)|.…

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Q: 6. Let f: A R be a continuous function. Show that if A is compact then sup f(A)e f(A).

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Q: b) If the function f(x), g(x) and h(x) are continuous on [a, b] and derivable on (a, b), show that…

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Q: |f(x) – f(xo)| < e whenever |x – xo] < 8. And the function f : R → R is continuous if it is…

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

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Q: 8. C'(0, 1] is the space of continuous functions f : [0, 1] → R with a continuous derivative f' :…

A: Let, fnn=1∞ be a Cauchy sequence in C10,1than for any ε&gt;0 there exist N∈ℕ such that ∀m,n&gt;N…

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: Suppose that f and g are continuous on I = [a, b], and they are differentiable on (a, b) with f'(x)…

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Q: Let f:Z x Z+ Z x Z be the function given by f(z, y)= (2z + y,I- y). Find, with proof, an element af…

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Q: Let (X, E, µ) be a measure space. Show that f is integrable if and only if fx ]f]dµ < o∞. Moreover…

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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…

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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…

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Q: Prove that f(2) = z/(z* + 1) is continuous at all points inside and on the unit circle Iz] = 1…

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Q: Define f(0,0) in a way that extends f to be continuous at the origin. 6x² -x²y² + 6y² x² +y? f(x,y)…

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Q: Show that the function f defined by 1, (x, y) = (1, – 1) f (x, y) : .2 x + y (x, y) # (1, – 1) x + y…

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Q: et f: (a, b) → R. If f'(x) exists for each x, a < x <b, prove that f is continuous there.

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Q: Show that a function f : R2-R defined by f(x,y)= x3- y3 / x2+ y2, if (x,y) is not equal to (0,0)…

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Q: metric

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Q: (b) Let f be a function defined by 1 if x e Qn [a,b] f(x) = - |-1 if x€ (÷) n [a, b]. Show that fis…

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Q: Let (X, T) and (Y, T1) be topological spaces and f a mapping of X into Y. If (X, T) is a discrete…

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Q: Let (X, d₁) and (Y, d₁) be metric spaces. Let f: X Y be continuous function, then f-¹(G) is open in…

A: Solution

Q: Find the Wronskian for the set of functions.{x, −sin x}

A: Given:  f(x1)=x and f(x2)=(-sinx ) The wronskian is given by the following determinant:

Q: Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x 0 а, f(x) = с, Is X with…

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Q: Let f:R" → R be a continuous function, and let x1, x2 € R" be distinct. Prove that there exist a…

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Q: Let f be Continuous on [a,b] and diffrentiable on (a, b), Prave that if azo, then there are Xi, Xz,…

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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: Let f:R→ R, be the identity function f (x)=x. Then f a) continuous but not open. b) open but not…

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Q: Let R have its usual topology and let Y = {x, y, z}. Define f: R → Y by: %3D x 0 (х, f(x) {y, (z,…

A: Quotient topology: If X is a topological space and A is set and if p:X→A is a surjective map, then…

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: