Q1- If f is bounded and define on [a,b] then show that m(b – a) < L(P,f)

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

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Q: Let f be a bounded function on [a,b] for which "f = ["f. Prove that f E R[a,b]. (Hint: Use the…

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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: 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: 4. Use the definition of uniformly continuity to prove that f (x) 1- x is uniformly continuous on…

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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: 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 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: 2. Let f : [a, b] R be a bounded function. Prove that if f is continuous at ro € [a, b), then L(|f])…

A: Here we use the property of continuty.

Q: 5. Let f : [0, 0) R be defined by f(x) = x+ 2 Prove that f is uniformly contin- %3D uous on (0, 00).

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Q: (b) Give an example of a bounded function f on [a,b] for which f2 e R[a,b), but f is not in R[a,b).

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Q: Theorem (6) If f is bounded and R – I on [a, b], f(x) > 0, Vx € [a, b] then ſ f 2 0

A: Prove that: If f is bounded and R-I on [a, b], fx≥0 for all x∈a, b , then ∫f≥0,

Q: 2. If f is continuous on an open interval containing x, and fchanges its direction of concavity at…

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Q: Prove that L(f) ≤ 0 ≤ U(f). Given that the bounded function f : [a, b] → R has the property that…

A: Given that the bounded function f : [a, b] → R has the property that f(x) = 0 for all x ∈ [a, b] ∩…

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: Let f be a bounded function on [a, b]. If P and Q are partitions of [a, b], prove that L( f , P) ≤…

A: Given f be a bounded function on a,b. If P and Q are the partitions of a,b. Then We have to prove…

Q: Let f,g : R → R be bounded and uniformly continuous. Prove that fg is also bounded and iniformly…

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Q: (a) Prove that, for every c E [0, 0), the equation xe" (b) Suppose f :R → R is a continuous function…

A: This is problem of Real Analysis, topology.

Q: 1 4. Let f be analytic on C. Prove that - 2n J0 \f(ret®)|d0 is bounded for 0 <r<∞ if and only if f…

A: To Prove :- 12π∫02πfreiθdθ is bounded for all 0≤r&lt;∞ if and only if f is bounded on ℂ. Given:- let…

Q: • Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum on (a, b).…

A: We have to give an example of a function f(x) that is not bounded on [a, b].

Q: 4. Prove fx=x2+4 is not uniformly continuous on R

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Q: Prove: If f is continuous on [a,b] and differentiable on nd f'(x) < 0, VE(a, b) then is ƒ is…

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Q: (b) If ƒ is continuous on (a, b) then there is a number c in (a, b) such that f(c) is an absolute…

A: Consider the statement  If f is continuous on (a,b) then there is a number c in  (a,b) such that…

Q: (5) Prove that Lp(f)<Uq(f) for any bounded function f and partitions P,Q of [a, b]. (Hint: Consider…

A: Note: According to Bartleby guidelines; for more than one question asked only the first one is to be…

Q: If (a) is decreasing and bounded below, then it is divergent. ylgn ihi

A: False

Q: The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that…

A: Consider the given 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: The function g is continuous on the interval [a, b] and is differentiable on (a, b). Suppose that…

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Q: 7. Prove that if the functions f and o are bounded and integrable in [a, b], then 6. 9. f(x) $(x) dx…

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Q: Q2. Show that F(x) = 3 – x², x € is contractive on its domain, but does not have any fixed points.…

A: To show - Show that Fx = 3 - x2, x ∈ -14, 14 is contractive on its domain, but does not have any…

Q: If f, g: R → R s.t. f and g are uniformly continuous. and bounded, then show that f.g is also…

A: Let f ,g are uniformly continuous And f and g are bounded

Q: 4. Let X = C[0, 1] be the space of continuous real valued functions on [0, 1]. Find a subset K of X…

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Q: Suppose that fn ⇒ f on R and that each fn is a bounded function. Prove that f is bounded.

A: Solution

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: If f, g : R R s.t. f and g are uniformly continuous. and bounded, then show that f g is also…

A: A differentiable function f:D→R is said to be uniformly continuous on D if for all ∈&gt;0 , ∃ δ&gt;0…

Q: 1. Use the definition to prove that the function in f(r) = 2r – 3 is uniformly continuous. 2. Use…

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Q: For all functions y = f(x) defined on [0,1], the set f([0,1]) is a bounded set. Is this statement…

A: For all functions y = f(x) defined on [0,1], the set f([0,1]) is a bounded set  Given statement is…

Q: Find an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) → 0…

A: To give an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) →…

Q: Let f be defined on [a, b], if f has a local maximum at a point x E (a, b), and if f'(x) exists,…

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Q: Consider the sequence of functions {f} defined on the interval [0, 1], where fn(r)= nx 1+n²r²¹ Vx €…

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Q: Suppose that f is integrable on [a,b], and that g(x) is defined for all x E[a,b] and ,exists and…

A: Given: To explain the given statement as follows, To prove that    exists and equals .

Q: The largest region, on which f(x,y,z)= 2-, is continuous, is y+1 x²+z²-2 O All points not on the…

A: According to question given thatf(x,y,z)=y+1x2+z2-2The largest region on which f(x,y,z) is…

Q: Let fi and f2 be concave functions from R" to R. Let f : R" +R be defined as the pointwise minimum…

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Q: 6. If af/ðæ is continuous on the rectangle R = {(t, x) : 0 0 such that |f(t, x1) – f(t, x2)| < K\x1…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

A: Let V=f:ℝ→ℝ. Let A is set of all polynomial with integer coefficient. We know that every polynomial…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 0 a limit…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.