Let I:= [a, b] and let f: I -> R be a continuous wetion such that f(x) >0 for each x iN I. Prove that here exists a Number of >0 such that f(x) > α for all XEI.

Q: Assume that a continuous function f:R → R is X- and µ-periodic, where A and u are two nonzero…

A: We know that between two rational number lies infinite many irrational number. But function is…

Q: Suppose that the function f:(a, b) → R is uniformly continuous. Prove that f: (a, b) → Ris bounded.

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Q: Let f be a real valued function defined on [a, b] that is differentiable on (a, b). Suppose f′(x) ≤…

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Q: Let I be an interval and let f : I → R be a strictly increasing function such that f(I) is an…

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Q: Let f be a continuous function defined on the closed interval [a,b] and differentiable on the open…

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Q: Let X be Hausdorff and f : X → X be a continuous function. Show that {x : f (x) = x} is closed in X.

A: Let, .We show  is open and so F is closed.

Q: 2. Suppose that f is an increasing function on the interval [a, b]. Let F(t) = %3D a.) Argue that if…

A: suppose that f is an increasing function on the interva a,b let Ft=∫0tfxdx (a) if a<x*<b then…

Q: A continuous function ƒ : [0, 1] → R satisfies ƒ(0) f(1). Show that for each integer n ≥ 1 there…

A: Let fix δ=1n, n∈ℕ Consider the continuous function gnx:0,1-δ→ℝ; gnx:=fx+1n-fx and suppose that gnx≠0…

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…

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: = √x + 2,-2 ≤ x ≤2 ax²-bx, x+1 x-2 ) 2 3

A: Topic:- calculus

Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

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Q: 12 Let u₁ = 1, ₂ = 1 and un + 2 = Un + 1 + un n ≥ 1. Then use mathematical induction to show that 1…

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Q: Suppose f: IR→ R and g: R → R and h: R→ R are continuous functions such that h(x) = f(x)g(x) for…

A: Using given information we have to tell that whether the  given statement is true or false.

Q: f : R → R+ is a function such that f(x y) = f(x) − f(y) for all x,y ∈ R+. Prove thefollowing (a) f…

A: Let the given function is defined as f (x + y) = f (x) – f (y). find the continuity of the function…

Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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Q: Let I be an interval and B a subset of R. Let f:I → B be a function continuous on I. AIM: To test…

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Q: • Assume f : A R continuous, and A open. Is f(A) open? • Assume f: A R continuous, and A compact. Is…

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Q: Suppose that f is continuous on [0,1], and that f(0)1. Prove that there is at least one point…

A: Intermediate value theorem: If f is a function that is continuous at every point of the interval a,…

Q: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

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Q: (b) Let X = (a, b]. Construct a continuous function f : X → R (set of real numbers) which is…

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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: Let f :ℝ--> ℝ continuous such that x - 1 ≤ f(x) ≤ x + 1 for all x ∈ ℝ . Show that for any y∈ ℝ there…

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Q: Suppose f and g are integrable functions on [a, b] and that for every interval [x, y] ⊂ [a, b] there…

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Q: 5.61737 = A + B (0.1936) 5.29414 - A A + 3. 114166 = A (0-1936)-c² B (0.2601) (0-2601)-C2 B(0-3721)…

A: We need to determine the values of A, B, C using the equations. A system of equations can be solved…

Q: t 0, if x is irrational 1, if x is rational ve a rigorous proof that f is not integrable on the…

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Q: Let f(x)=1/x. Show that f is bounded from above on any interval (α,1), where α>0, but the function…

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Q: A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1…

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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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