1. Show that the function, defined for x E R by f(x) = 1 is uniformly continuous 1+x² on R.

Q: (0,1). Prove the following functions is uniformly continuous on sin x f(x) =

A: Uniformly Continuous:-  Here we prove the result using a theorem of uniformly continuous.

Q: Let f(x) be a continuous function satisfying f(x) = (x² + 1) || f(t) dt + 2x. Compute f(0.9)…

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Q: Theorem 7. Let X be a continuous RV, g a function from R to R. Let Y = g(X). Then E[Y] = E[g(X)] = /…

A: The expected value of random variable is the weighted average of all possible values of the…

Q: Define h(2) in a way that extends h(t) = (t2 + 3t - 10)/(t - 2) to be continuous at t = 2.

A: Consider the given the function,

Q: Let f(x) be a continuous function on R-{0} such that (f(x) dx exists for all a > 0.If g(x)= |- dt, t…

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Q: 2. Let f(x) = S dt for x 2 2 In a) Find F'(x). b) Is F increasing or decreasing? What can you say…

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Q: 4. Choose and work either a) or b), NOT BOTH. (a) Show that the function f (x) = sin z for x €R is…

A: In the question given that we have to solve any one of them. Let us solve the first one.

Q: c. Show that f(x) is NOT uniformly continuous on [0, +∞).

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Q: 2. Show that the function, defined by f(x) = is NOT uniformly continuous on its x2 %3D domain D(f) =…

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Q: 2. If f is a continuous function such that S(t)dt = cos(2x)e* - [e s(t)dt for all x, find an…

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Q: Prove that f(x) =√x is uniformly continuous on [0,∞).

A: According to the question, we have to prove that the function fx=x is uniformly continuous on [0,∞).…

Q: iii. is continuous at a provided g(a) # 0.

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Q: Consider the function xy f(x, y) x²+xy+y² k if (x, y) = (0,0) if (x, y) = (0,0) State, with…

A: Given :  f(x, y) = xyx2+xy+y2if (x, y)≠(0, 0)kif (x, y) = (0, 0) To find : The value of k if it…

Q: [0, 4). 4. Prove that f(x) = x² is both uniformly continuous and continuous on

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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: B) Let f:[-2,2]R. then show that the function xsin -) for x 0, f(x) = for x = 0, is uniformly…

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Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

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Q: Given a continuous function of which is twice differentiable, if f" (c) > 0, which of the following…

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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: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

Q: Prove f(x) = x2+4 is not uniformly continuous on R.

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Q: If f(x) is differentiable in (a, b) and there is M > 0 such that |f'(x)| < M for all x E (a, b),…

A: Topic = Functions True

Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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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: Use the e- 8 property in Definition 19.1 to show that f(x) 1 is uniformly continuous on the interval…

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Q: how (in terms of  − δ) that a function f : R 3 → R defined by f(x, y, z) = (2x + 3y + 4z) is…

A: Solution of given problem is followed by

Q: If x is ar.v. with c.d.f. F(x), F(x) is continuous at x = 3, then p(x = 3) = 0 %3D O True False

A: It is given that the CDF of x at x = 3 is continuous.

Q: Let X ~ N(H, o²) and suppose h(x) is a smooth bounded function, xE R. Prove Stein's formula E[(X –…

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Q: 1. Show (in terms of  ε - δ) that a function f : R3 → R defined by f(x; y; z) = (2x + 3y + 4z) is…

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Q: 4. If L is a Continuous function on H, then unique YEH such that L(x)=(x,y). уєн there is a

A: Given that L is continuous on H. To show: y is unique in H.

Q: : Let g : R → R define by 3x for x E Q Then g(x) 2x – 3 for x € Q' :the point where g is continuous…

A: Let f be a function. The function f is said to be continuous at the point c if it satisfies the…

Q: Exercise 0.4. Assume that g : R → R is a periodic continuous function. Prove that f(x) = 2x + g(x)…

A: Continuity and uniform continuity are two different things. A continuous function may not be…

Q: (a) Prove that if m ₹ N, then the function f(x) = xm is continuous on R.

A: We have to prove that, the function f(x) = xm,   m∈ℕ  is continuous on ℝ.

Q: 2. Suppose that f is a measurable function satisfying m({r €R: [f(x)| 2n}) < ne N. n3' Is f…

A: Given that f is a measurable function, with the following condition satisfied. m(x∈ℝ:…

Q: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

Q: Show (in terms of  −δ) that a function f : [2,7] −→R be defined by f(x) = √x2+1 is uniformly…

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Q: Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is…

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Q: Let f(x) x² and g(x) = xf(x). Find c = (0, 1) as guaranteed by the Cauchy Mean Value Theorem.

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Q: 1. Show (in terms of -5) that a function f: [2, 7] R be defined by f(r) V+1 is uniformly continuous.

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Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

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Q: Determine if f(x)=In (x² + 1) is uniformly continous on the interval I (-0, +0)

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Q: Show that f : (0,∞) → R defined by f(x) := 1/x is not Lipschitz continuous.

A: Definition:  Let S⊂ℝ and a  function f:S→ℝ is Lipschitz Continuous at x∈S if there is a constant C…

Q: 1. Suppose f(r) is a continuous function on r € [a, b]. Prove f(x)dr < S°S(x)| dx.

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Q: 1. Estimate the limit of the signum function s(x) as x approaches zero from the right. (-1, x0

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Q: 2. Determine whether the following functions are linearly dependent or not by applying the…

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Q: Let f(x) = x² and g(x) = xf(x). Find c = (0,1) as guaranteed by the Cauchy Mean Value Theorem.

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Q: 2. Suppose that f is a measurable function satisfying m{r ER: |f (x)| > n} ) < 1 nEN. n3 Is f…

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Q: Show that f(x) = 1/x2 is uniformly continuous on the set [1,∞) but not on the set (0, 1].

A: To show that f(x)=1x2 is uniformly continuous on the set [1,∞) but not on (0,1].