Prove: If S is compact, and ƒ is continuous on S, then ƒ takes a minimum value some- where in S.

Q: .4 || (cos²e + sin?e) dp dz de 8 2(0*cos?8 + p*sin?e) p dp dz de

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Q: Let (X,F) be a measurable space and let f, g: X → Y be F -measurable functions. If E E F, show that…

A: The given in this question is a measurable space (X,F) and two F-measurable functions f and g from X…

Q: Suppose ƒ is continuous on [1, 5] and 2 ≤ƒ(x) ≤ 3, for all x in[1, 5]. Find lower and upper bounds…

A: If f:[a,b]-->R is a continuous function then f is integrable over [a,b].

Q: Prove: Let the function f:X →Y and g:Y →Z be continuous. Then the composition function g•f:X→Z is…

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Q: Let X be a normed space and let 0x0 EX. Show that there is f = X* such that f(x) = 1 and ||f|| =…

A: Given, a normed space  and .It is required to show that, there exists  (the dual space of ) such…

Q: -> 2. Prove: Suppose that a function f: [a, b] → R is bounded and P = {a = x0, x1, x2, ..., xn = b}…

A: The objective of this question is to prove that for a bounded function f defined on a closed…

Q: 44. Suppose that ƒ : [a, b] → [a, b] is continuous. Prove that there is at least one fixed point in…

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Q: Px (w) is a continuous function of w in (- ∞, 0) for any real value of ow, and in fact, it is…

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Q: Let h: R -> R be a uniformly continuous function on R. Show that there exist positive constants a…

A: Let   be  a  uniformly  continuous  function  on We  have  to show  that  there  exists  positive…

Q: Is the proof correct? Either state that it is, or circle the first error and explain what is…

A: The given proof is correct.

Q: D. Prove that f(x) = |x - 6| is continuous at all points c in R.

A: Given function is fx=x−6. We have to prove that the function fx=x−6 is continuous at all point c in…

Q: 3. Let f, g : [a, b] → [c, d] be cotinuous functions. Suppose that f is surjective. Prove that there…

A: We will use of intermediate value theorem to proof this

Q: Let DCR and let f: D→ R be a function. For a set B CR the preimage of B under ƒ is f−¹(B) = {x € D |…

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Q: If f is continuous on [a, b], then the horizontal line y = k for every k between f (a) and f (b)…

A: If function 'f'  is continuous on closed interval [a, b], and if f(a)≠ f(b), then for any number k…

Q: Let f : E → R be uniformly continuous. Prove that |f| is uniformly continuous. Is the converse true,…

A: Let f: E → R be uniformly continuous.  Now, we need to show that |f| is uniformly continuous.  Now,…

Q: (a) Assume f: S → R is uniformly continuous and S is bounded. Prove that the range of f is bounded.…

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Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

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Q: Let a < b be given. Give examples of continuous functions g, h : (a, b) → R such that (a) g is not…

A: Let a&lt;b be given.We need find the examples of continuous functions g,h:(a,b)→R such that:(a) g is…

Q: 4. Let A C R" be both connected and sequentially compact. Prove that if F: A –→ R is continuous,…

A: Consider the given information.

Q: Let n > 2 be a natural number. Prove that there is no continuous function q : C → C such that…

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Q: A function f continuous on (a, b) always has a relative maximum and a relative minimum on [a, b]…

A: We have to check

Q: Let S C C be a domain that is bounded, and let f: SC be continuous, and f = O(U) is holomorphic. If…

A: To prove that f must have a zero in S, it can use the Maximum Modulus Principle. The Maximum Modulus…

Q: example of a function f:[0,1]→ℝ such that f is not continuous, but f does satisfy the conclusions of…

A: Given function f:01→R f is not continuous, but f satisfies the conclusion of extreme value theorem

Q: Prove that if S(c +h) – f(c) lim h exists, then f is continuous at c.

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Q: (A) For each metric space 2 and A below determine whether there exists a continuous function f onto…

A: We know  A function(say f:X =&gt; Y) is continuous if and only if the  inverse image of every open…

Q: 2. Is the proof correct? Either state that it is, or circle the first error and explain what is…

A: The proof is, Let ε&gt;0 Since f is  uniformly continuous on (0,1] there exists δ1&gt;0 and…

Q: Given the function f:{1,2,3} → N defined by f = {(1,3), (2,5), (3,1)} Is f1-1 Is fonto What is the…

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Q: Prove: If ƒ e R(a) on [a,b] and g e R(a) on [a, b] , then fg e R(a) on [a,b]

A: Given if f∈Rα on a,b and g∈Rα on a,b. We have to prove that fg∈Rα on a,b. Since, f∈Rα on a,b then f…

Q: Exercise 3.3.10: Suppose f: [0,1] → [0,1] is continuous. Show that f has a fixed point, in other…

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Q: For matrix A = (4 6 3 5 ). If f(x) = x2 - 5, calculate f(A)?

A: The given matrix is, A=4635. The function is f(x)=x2-5. For matrix multiplication: The number of…

Q: Let f : [0, 1] → [0, 1] be continuous. Show that there exists an element x = [0, 1] such that f(x) =…

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Q: 3 If fix SY compact, then so is f(x). continuous and X is

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Q: etermine the equation of your model that will pass 2,4

A: We already have the equation of polynomialand

Q: Prove that f: R→ R is continuous. 2. Suppose that the function f : [0, 1] → R is continuous and that…

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Q: Let f RR be continuous. If f(c) > 0, prove there exists a neighborhood N of c : such that f(x) > 0…

A: Let f: R -&gt; R be "continuous''.Claim: If f(c) &gt; 0,            prove there exists a…

Q: Let f(x) be a given continuous function on a closed interval [a, b]. Use the Extreme Value Theorem…

A: given fx is a continuous function on a closed interval a,b to prove- use extreme value theorem to…