Suppose f : [a,∞) → R is continuous and f(x) → 0 as x →∞. Prove that f is uniformly continuous on [a,∞).

Q: Suppose DCR and f.D R and g:D-R are both uniformly continuous on D. (a) Prove that the function f +…

A: Uniform continuous function: a function is called uniformly continuous if for every real number…

Q: Prove that f(z)= 1/(1-z) in not uniformly continuous for |z|

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Q: Let BCR be the open ball of radius one centered at the origin, i.e. B = {(x₁,...,xn) € R" : x² + +…

A: As per the question we are given a n dimentional open ball B of radius 1 centered at the origin. And…

Q: 24. Show that f(x) = mx + b, where m,b e R, is uniformly continuous on R. %3D

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Q: If f(x) is uniformly continuous on R, then f is bounded on R. True False

A: To say True or False

Q: Let f(x) = x sin x. a. Compute f'(x). b. Show that for every M > 0 and every k > 1, there is xo > M…

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Q: Let g be bounded on [a, b] and assume there exists a partition P with L(g,P) = U(g,P). Describe g.…

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Q: If f is continuous on [a, b], then there is cE[a, b] such that f(c) = av(f). Select one: True False

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Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

A: Let f:X,Tx→Y,Ty be a function and a is open in Tx. To prove that the function is continues at a.…

Q: Let a, b, c e R with a < b < c. Suppose that f is uniformly continuous on [a, b] as well as on [b,…

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Q: Prove that if f: A to R is continuous at x=c, then for all (x_n) converging to c, it follows that…

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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: Suppose that f is continuous on [0, 2] and that f(0) x, y in [0, 2] such that|y – x| = 1 and f(x) =…

A: If is given that f is continuous on 0, 2 and f0=f2. To prove: There exists x, y∈0, 2 such that y-x=1…

Q: Suppose that f is a function on [a, b] such that f(x) = 0 for all x e [a, b]\T , where T is finite…

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Q: Let f : R → R be a function such that |f (x) – f(y)| 0. Write the e-d definition of continuous…

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Q: Let X be a set, let fn : X –→ R be a sequence of functions, and let f : X → Y be a function. Let ē…

A: given let X be a set , let fn:X→ℝ be a sequence of function and let f:X→Y be a function. let ρ¯ be…

Q: Let f: X -> Y and g: Y -> Z be uniformly continuous on X and Y, respectively.  Prove g ◦ f: X -> Y…

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Q: Let f be continuous on [a, b] and suppose that f(x) 2 0 for all x e [a,b]. Prove that if L(f)=0,…

A: Given: Let f be continuous on [a,b] and suppose that f(x) is greater than or equal to zero for all x…

Q: Prove that, if f is differentiable on (a,b) and f′ is bounded on (a,b) then f is uniformly…

A: Applying mean value theorem,

Q: 3. Suppose f, g:D → R continuous on D. are uniformly continuous on D. Prove that f + g is uniformly

A: Since you have submitted more than one question so according to our policies we can just answer one…

Q: Prove that if f, g :R → R are continuous at x E R, then ƒ+gis continuous at x E R as well.

A: By using the definition of continuous function this above statement can be easily proved.

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: 4. Let f : X → Y and g : Y → Z be uniformly continuous on X and Y, respectively. Prove gof: X → Z is…

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Q: Let f:X→R be a Lipschitz continuous function defined on X. Show that f is uniformly continuous on X.

A: Given : function f:X -&gt; R is a Lipschitz continuous function defined on X . To show : f is…

Q: Let X be a set, let fn : X → R be a sequence of functions, and let f : X → Y be a function. Let ē…

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Q: Let f, (x) uniformly converge to f(x) on [a, b]. If fn(x) E R[a, b], then f(x) E R[a, b] False True

A: We have to find correct statement.

Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

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Q: . If f is a continuous function from the metric space X to the metric space Y and {xn} is a sequence…

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Q: If f satisfies a Lipschitz condition on ICR, then f is uniformly continuous on I. a.

A: Since you have sent more than one question, because of our guidelines we can just answer one of…

Q: Let f be the function defined by f(x) = . Use the definition of continuity of a fuction to show that…

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Q: Assume that g : R → R is a periodic continuous function. Prove that f(x) = 2x + g(x) is uniformly…

A: The complete solutions are given below

Q: Let f: X → Y and g : Y Z be uniformly continuous on X and Y, Prove go f: X→ Z is uniformly…

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Q: Suppose that f and g are continuous on I = [a, b], and they are differentiable on (a, b) with f'(x)…

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Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

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Q: Use the definition of uniform continuity (i.e., the e, d definition) to prove that f(x) = x² – x is…

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Q: Suppose f is uniformly continuous on (a, m] and also on (m, b). Prove: f is 1 uniformly continuous…

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Q: Suppose f,g: D -> R are uniformly continuous on D. Prove that f+g is uniformly continuous on D.

A: To Prove- Suppose f, g : D → ℝ are uniformly ccontinuous on D. Prove that f + g is uniformly…

Q: et f: (a, b) → R. If f'(x) exists for each x, a < x <b, prove that f is continuous there.

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Q: For a metric space (X, d) and a continuous functionf from X into itself, show that the set of points…

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Q: Let f be a continuous function on [a, b] such that f(r)a" dx = 0 for every n > 0. Prove that f = 0…

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Q: Suppose f is uniformly continuous on [n, n + 1] ∀ n ∈ N. Does it follow that f is uniformly…

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Q: Let f, (x) uniformly converge to f(x) on [a, b). If f.(x) e R[a, b), then f(x) € R[a, b] True False

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Q: 3. Let f, g : D →R be continuous at c € D. Prove that fg is continuous at c.

A: Since you have asked multiple questions, we have answered the first question. If you want the second…

Q: 2. Use the definition of uniform continuity to show that any Lipschitz function f on E is uni-…

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Q: Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] and on [1,∞). Show that it is…

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Q: Let a, b ∈ R with a < b. Show that a function f : (a, b) → R is uniformly continuous on (a, b) if…

A: Let fx is uniformly continuous in a,b We have to show that f can be extended to continuous on a,b.…

Q: Suppose f : (M, d) → (N, p) is continuous. Prove or disprove: f is uniformly continuous + f maps a…

A: f: (M,d)→(N,ρ) Proof: fis uniformly continuous ⇔ f maps a cauchy seq. to a cauchy seq.

Q: Assume that f is continuous in [a, b], twice derivable in (a, b), and that f (a) = f (d) = f (b) = 0…

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Q: Let f,g: D -> R be conitnuous at c ∈ D. Prove that fg is continuous at c.

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.