Let KCR be compact and f: K→ R. Suppose that for each x E K there exists x > 0 such that f is bounded on Kn N₂(x). Prove that f is bounded on K.

Q: 3. If f : R R is continuous at ro E R and f(ro) > 0, prove that there exists an interval containing…

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Q: Define f : R → R by f(x) = 0 if x∈ R\Q f(x)=1/q if x=p/q with (p, q) = 1. Prove that f is…

A: Define f:ℝ→ℝ byf(x)=0if x∈ℝ\ℚ1qif x=pq with (p,q)=1Prove that f is continuous on ℝ\ℚ and…

Q: f: R -> R be contnuous. If f(c) > 0, prove there exists a neighborhood N of c such that f(x) > 0 for…

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Q: Suppose f: [a, b] → R is a bounded function. True or false: If f is Riemann integrable, then for…

A: Since U(p, f) is the upper bound of the integral. And the given condition is the the condition of…

Q: (a) If ƒ and g are integrable, then gof is integrable. (b) If ƒ is increasing and g is integrable,…

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Q: Let f : E → R be continuous such that |f(x)|≤|x|, for all x ∈ E. Is f uniformly continuous, justify

A: According to the given information,

Q: Letf:E→R be uniformly continuous. (a) Prove that E is bounded⇒f(E) is bounded.

A: Given, f:E→ℝ is uniformly continuous. Since, E is a bounded subset of ℝ and f is uniformly…

Q: Let f:R→R. If f is defined b then it is Cantindous. foo=2x+1 if X>o if X5o

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Q: Let f : D → R be continuous at c ∈ D and suppose that f(c) > 0. Prove that there exists an α > 0 and…

A: To prove this statement, we can use the definition of continuity.

Q: Let f, g : R → R be continuous at 0. Using only the e – 8 property(Ross, Theorem 17.2), prove that…

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Q: Q3 Prove that if f and g are continuous at zo in R, then min(f, g) is continuous at 0.

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Q: Let S be a closed n-cube with v(S) > 0 and suppose f : S –→ R is continuous on S (relative to S).…

A: let S be a closed n-cube with vS>0 and suppose f:S→R is continuous on S (a) prove that infSf and…

Q: Let f be defined on an interval I. Suppose that there exists m > 0 and a> 0 such that: If (x) -…

A: Let us assume "A" is a  non empty subset of R, then f:E→R is uniformly continuous if and only if for…

Q: Suppose that a, b R with a 0. Prove that [sz f≥ 0.

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Q: Suppose f is differentiable on (0,1) and that there exists M > 0 so that |f′(x)|≤M for all x ∈(0,1).…

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Q: . Let f be a bounded function on [0, 1], say |ƒ(x)| ≤ M for all x € [0, 1]. Show that all the…

A: To show that all the Bernstein polynomials Bnf are bounded by M, will first define the Bernstein…

Q: Suppose f: R→→ R is convex. Show that for all x € [a, b]. x-a :) ≤ b = = f(a) + ² = f(b) b a f(x) ≤

A: Suppose   is convex. We have to show that:

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: Suppose the function f : R → R is differentiable at every x ER and assume that f' is bounded (so…

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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: Exercise 10. Show that if ACB, then ACB.

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Q: Assume f : E → R is continuous at a ∈ E and f(a) 6= 0. Show that there exists an open interval I…

A: According to the given information,

Q: 3. Use the Bolzano Weierstrass Theorem to prove that if f is a continuous function on [a, b), then f…

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Q: 1. Construct a continuous function f: I R on a bounded interval I with the following property: f is…

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Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

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Q: * 7. Suppose g : [a,b] 8 E R(x) on [a, b). See Exercises 24 and 25 for generalizations of this…

A: Given - Suppose g : a, b → R is continuous except at x0 ∈ a, b and bounded. To Prove - Prove that…

Q: Let f : R→R be a continuous function such that for every x, y∈R, f(x+y) =f(x) +f(y). Prove there…

A: Given that   f:ℝ→ℝ be a continuous function such that                                       for…