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

Q: if x 4.

A: The function gx is a piecewise defined function.

Q: Suppose that the function f : R → R is continuous for x ≥ 0, differentiable for x > 0, and satisfies…

A: Mean Value Theorem Suppose  f ( x )  is a function that satisfies both of the following conditions.…

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: Show that a uniformly continuous function on (a, b) is bounded.

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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: A function f : D → R is called a Lipschitz function if there is a number K > 0 such that |f(x) –…

A: Lipchitz function is uniform continuous on the given domain. This proves that f is uniformly…

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 E = {0, 2, 4, 6, ...}, define ƒ: E × E → Z¹ by the formula f(m, n) = 4m5¹, \ (m, n) € Ɛ × Ɛ.…

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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: 3. Assume f: ZR and prove that f is uniformly continuous.

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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: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

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: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

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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: (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 => 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>0 Since f is  uniformly continuous on (0,1] there exists δ1>0 and…

Q: Suppose that f and g are both continuous functions. Also f, g € [a, b] and that f= fg. Prove that 3c…

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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: (iii) (f/g)(x) = g(x) if g(x) #0 for all z € M. Prove that cf. fg and (f/g) are all continuous. 8.…

A: To Prove: Restriction of a continuous function is also continuous on Metric space.

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: 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 A := (0, 1] and let ƒ : A → R be defined by f(x) = !. Prove that ƒ is continuous on A.

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