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

Q: Let f: A→B be a function. Prove or disprove the equivalence: (a) f is injective (b) If C is any set…

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Q: What is the global minimum of the function, x² 5 f(x) 3 + 2 on the interval [-2, 2] ? a) 2 b) 23 3…

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Q: Does there exist a set of all functions? Suppose such a set exists and call it F. Define a…

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Q: Prove that the Cantor function is not differentiable on the Cantor set.

A: The complete solutions are given below

Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

Q: Prove that a continuous real-valued function on a compact set takes on its absolute maximum and…

A: It is to be proved that, a continuous real valued function on a compact set takes on its absolute…

Q: 15. Let A be an infinite set. Prove that there exists an injective function f: N→ A.

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Q: Find example where of is continuous and an - bijective but f¹ is not

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Q: fa) = b, Ab) = a, c) = C, 1d) = d is a one-to-one function, as each element is an image of exactly…

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Q: Prove the Theorem: A real valued function on a closed, bounded interval is continuous iff its graph…

A: If we assume that f(x) is unbounded at c , then bisecting the interval [a,b] would give us two…

Q: Suppose both f(z) and the conjugate of f(z) are holomorphic on a connected open set.  Show that f is…

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Q: Define a bijective map g : x2/3--->R.

A: Sol

Q: Theorem 13.6 Consider the function f:A – B. Let Uj and Uz be arbitrary subsets of the domain A, and…

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Q: Let f:X → Y be a linear function, where X and Y are normed spaces over the field of real numbers.…

A: The given linear function: Where X and Y are normed spaces over the field of Real numbers. To prove…

Q: A function, F: R × R ---› R x R has been defined as follows: F(x, y) = (3y - 1, 1 - x) for all (x,…

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Q: Given that f is a function from A onto B, that g :B →C, and that the function g • f is one-one,…

A: In the given question we have to find f and g are one - one function.

Q: Assume (r1, y1) ~ (x2, Y2) → f(x1,Y1) = f(x2, Y2). Prove that ~ an equivalence relation.

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Q: Prove that if f: R→ R is continuous, then the image of a bounded set over f is bounded.

A: As per the question we have to prove that if any function f : ℝ → ℝ is continuous, then the image of…

Q: Let g : X c R"→ R be a function defined on the finite set X. Does g achieve a maximum and/or a…

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Q: if (X,d) is discrete metric space, ACX and A+D (x,d) Find the sets of A° and A (functional analysis…

A: Consider a discrete metric space X,d where d is defined as: dp,q=0,p=q1,p≠q

Q: Provide an example of function f: R → R which is continuous at exactly one point and discontinuous…

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Q: Let f be the function from R to R with f(x) = 3x - 5. a) Prove in detail that f is one-to-one. b)…

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Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: Exercise 3 Prove that a function f: Rd - Rm is continuous if and only if f¹(U) C Rd is open for any…

A: Given:A function , where  represents the -dimensional Euclidean space and  represents the…

Q: Define the relation A=((x,x+ 1):xER] Show that A, is a function from R to R.

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Q: fa function f is continuous on the closed interval a, b, a, b E R, then f has an absolute maximum…

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Q: Let A={1, 2, 3, 4, 5} and B={a, e, i, o, u}. If ƒ: B → A is a function defined by f ={(a, 4), (u,…

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Q: Let fand g be the functions defined by f(x)= x+et-&x + 1, and g(x) ==- 4x + 2. Let Q and R be the…

A: Hi! All the sub-parts are quite big. As nothing is specified, we are solving (a). kindly re-post the…

Q: Let X be the set {5, –6, 2}. For the first three parts of this problem you are asked to define a…

A: Consider the set 5,-6,2. The function is f:X→X and the relation is u~w⇔w=fu.   a) The relation is…

Q: Let f : X → Y be a function. Define a relation R in X given byR = {(a, b): f(a) = f(b)}. Examine…

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Q: Let f : R → R and ECR. Prove f is continuous iff f-¹(E) is closed whenever ECR is closed.

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Q: If f is a function from X to Y and A, B are subsets of X then .A f(A – B) = f(A) – f(B) .B f(AUB) =…

A: Given: f:X→Y and a,b⊆X, Find out: To choose the correct option.

Q: Give an example of a function R -> R that is discontinuous with the usual topology but is continuous…

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