Decide which of the following conjectures is true and supply a short proof. For those that are not true, give a counterexample. (a) If |f| is integrable on [a, b], then f is also integrable on this set. (b) Assume g is integrable and g(x) ≥ 0 on [a, b]. If g(x) > 0 for an infinite number of points x ∈ [a, b], then ) b a g > 0.(c) If g is continuous on [a, b] and g(x) ≥ 0 with g(y0) > 0 for at least one point y0 ∈ [a, b], then ) b a g > 0.

Q: interval [a,b] where it is NOT integrable. f(x)=           ,   x∈

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Q: If f is bounded and integrable on [a, b] such that | f(x) | 0).

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Q: 2. Show that if f is continuous on [a, b] with f(x) > 0 for all a ≤ x ≤ b, then 1/f is bounded on…

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Q: Let f : R → R be the function defined as if x 1. Use Theorem 32.5 to prove that f is Darboux…

A: A function is Darboux integrable if for all ε>0, there exists a partition P of [0,2] such that…

Q: Suppose that the function f is continuous and strictly monotone on the interval [a; b]. Choose the…

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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: Define f: [0, 1] →→ R by the property that 1, if is rational, f(x) = a 0, if ar is irrational. True…

A: f:0, 1→ℝfx=1, if x is rational,0, if x is irrational.

Q: 3. Let f : [2,5] → R such that f(x) et Is f uniformly continuous? Justify your answer. Answer:

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Q: Let f: [a, b] -> R be continuous except on a finite number of points in (a, b]. Is f Lebesgue…

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Q: Consider the function f : defined by                    f(x) = 0 if x is irrational and f(x)=1 if x…

A: The objective of the question is to determine whether the given function f, which is defined…

Q: Let f : R" → R be continuous. Prove or give a counter example: if O C R" is open, then f(0) is open.

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Q: (d) Let A = D(Rº) and suppose that ƒ is integrable on A. Show that |ƒ| is also integrable on A and €…

A: Let A belongs to D(Rp)  Suppose f is integrable on A Let [a,b] belongs to A

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

A: Topic:- calculus

Q: Let f : [a, b] → R be increasing on the set [a, b] (i.e., f(x) ≤ f(y) whenever x < y). Show that…

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Q: Let a, b € R, a < b. Let f be an integrable function on [a,b), let c E (a, b), and let F(x) = |…

A: Solution for a: If f is differentiable at c, then F is differentiable at c Recall: First Fundamental…

Q: Let f : R" → R be continuous. Prove or give a counter example: if O c R" is open, then f(O) is open.

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Q: Let f : R" → R be continuous. Prove or give a counter example: if O c R" is open, then f(0) is open.

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Q: Prove that the function f(x) = Root3(x) is uniformly continuous on [−1, ∞). (Hint: consider the sets…

A: Given that fx=3x we have to prove that fx is continuous in interval [-1,∞) lets consider the set…

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

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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&gt;0 and suppose f:S→R is continuous on S (a) prove that infSf and…

Q: Assume the function f(x, t) is continuous on the rectangle D = {(x, t) : a ≤ x ≤ b, c ≤ t ≤ d}.…

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Q: Assume f : [a, b] → R is integrable. (a) Show that if g satisfies g(x) = f(x) for all but a finite…

A: We know that the function f is integrable if f is continuous everywhere in the interval including…

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: If f is bounded and integrable on [a, b] such that | f(x) | 0).

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Q: Prove that the function f(r) = is uniformly continuous on (0, 00) エ十2

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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: Consider the transformation T:R'- R defined by T(1, y, 2) = (-21 +z, 4y + 32) (a) Prove that T is a…

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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: 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,

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.