2. Let f and g both be of bounded variation on [a, b]. Prove that this inequality holds. Vrag SV+V

Q: B. f(t) = { * ' 5, t + 1, 0<t<2 %3D 2sts 10

A: given f(t)=t+10≤t&lt;252≤t≤10

Q: 16.S. If f is uniformly continuous on a bounded subset B of RP and has values in Rº, then must f be…

A: Given that, fx:a,b→ℝ is uniformly continuous. We will prove this by contradiction. So, fx:a,b→ℝ is…

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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: 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: 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…

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Q: Which of the following is at least piecewise continuous in the interval [0, 10]. 2t, 0 st<1 2t - 3 ,…

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Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

A:

Q: 3. Use the e, d definition to show that f(x) = x² is uniformly continuous on A = [0,3]. %3D

A:

Q: 1. Show that if f [a, b] → R is increasing, then f is of bounded : variation.

A:

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: . Apply Jensen's inequality (2) to f(x) = In x, with i 1 = ..= d = 1/n to prove that 1 Vx1x2...X 0,…

A: Given :-  Apply Jensen's inequality (2) to f  x  = ln x  , with λ1 = λ2 = ..........  = λn = 1n to…

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 : 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: 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: Derive a finite difference approximation to of(x,y) + f(x,y) using the points f(x, y), ƒ (x+h, y),…

A: Here we use taylor series to find this.

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: Prove that if f.(x), g,(x) converges uniformly on a set A and are uniformly bounded on A then f,(x)…

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Q: Suppose that f is a continuous function satisfying stæ) = a f(t)dt + * So f(t)dt + a %3D for all a,…

A: Solve the function

Q: 1. Show that if f [a,b] → R is increasing, then f is of bounded variation.

A: Given f: a, b → ℝ is increasing function.

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: Let f (x) be a nonconstant element in Z[x]. Prove that k f (x)l is notmaximal in Z[x].

A:

Q: Consider the ball B1(0) of R°. Any function of Wi3 (Bo(1)) is uniformly bounded up to null sets. O…

A: Note: the ball B1(0) is an open ball in R3

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

A:

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: Let F subset of K , where dim_F(K)=2. Show that K is normal over F . I Need typed solution…

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

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: 8. PROVE: Suppose that SCR is connected, and that f : R → R is continuous. Then f(S) is connected.…

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Q: 2. (a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if…

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

Q: Prove: If S is compact, and f is continuous on S, then f takes a minimum value some- where in S.

A: Since S is compact and f is continuous, f[S] is a compact subset of R and therefore closed and…

Q: Use the definition of uniform contimuity to prove that (r) is uniformly cotinncs (0, x)

A:

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

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Q: Suppose that f:[a,b] → R is a bounded function for which there sa partition P of [a, b] with L(f,P)…

A: Given the statement Suppose that f: [a, b] → R is a bounded function for which there is a partition…

Q: Let f(a) - Compute o o+CIT for the domain 0 aS1. Let ro and ny . %3D • Compute the continuous…

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Q: Suppose f,9 : D R are continuous on D. uniformly continuous on D. Prove that f +g is uniformly

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Q: Use Hölder inequality to prove l|u|la < llu||||;- for u E L" (N) 1 1-2 Where p <q <r and =

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Q: Suppose f is differentiable on (a, b). If c E (a, b) and f' (c) = 0, thenf (c) is either the maximum…

A: Given f is differentiable on (a,b). and c lies in (a,b) such that f'(c)=0.

Q: Let f(x) = 7sin x for 0 sxs . Compute Lf (P) and Uf (P) for the function f and the partition P = Lf…

A:

Q: Which of the following is at least piecewise continuous in the interval [0, 10]. 2t, 0st<1 2t - 3 ,…

A: A)This function  is not defined for [2,5) so it is not piecewise continuous on [0,10]   B) this…

Q: Prove that f : [−4,2] → R defined by f(x) = 7/x+5 is uniformly continuous on the interval [−4, 2]…

A:

Q: 3. If f is continuous and | f(2) dr = 10, find | af(x*) dr. %3D Give proper reasoning.

A: It is given that if f is continuous and ∫09f(x)dx=10 We have to find ∫03xf(x2)dx=?

Q: Let f: X→Y, g:Y-Z be uniformly continuous. Then gof:X-Z is uniformly continuous.

A: The main objective is to proof that the function g∘f is uniformly continuous, i.e. for every ε&gt;0,…

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: (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: Assume that is a continuous real function on (a, b) such that 1 1 o ( x + ³) ≤ ²}/20 (x) + = 0 ( for…

A: Here given that φ is a continuous real function on a, b such that φx+y2≤12φx+12φx

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: Exercise 11. Prove that if f: E R is continuous at p E E, then there exists & >0 so that f(x) is…

A: