: 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 is 1 0 3 0 2 0 -3 O

Q: 33. Let x² - 4y2 if x + 2y f(x, y) = 2y %3D | 8(x), if x = 2y %3D If fis continuous in the whole…

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Q: 3. If f : R R is continuous at ro E R and f(ro) > 0, prove that there exists an interval containing…

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Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

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: Let f(r, y, z) = In(r2 + 2) + y?e* – cos(2yz). Find Szya- Find all the relative extrema and saddle…

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Q: Consider the function 2+2+z9-2ry+y if (r,y) # (0,0) f(1, y) = if (1, y) = (0,0). (a). Find and at…

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Q: { z sin a). if (x, y) # (0,0) 2. Determine if a function f: R2 R defined by (. 9) = BE 0, if (KJ) =…

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Q: Let f : [0, 1] → R be a continuous function. Show that the function g : [0, 1] → R given by g(x) =…

A: The given function: is continuous. That is, assume f(x)=x, then x is Real number. Show that the…

Q: Assuming that z is a differentiable function of x and y, Let F(x.y.2) = In(x³y) – e" -tan-z. Find F,…

A: We can find the answer as below.

Q: 1. Show (in terms of e – 8) that a function f : R³ → R defined by f (x, y, z) = (2x + 3y + 4z) is…

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Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(T)<0.

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Q: 2. Define, for x E S [0,1] U [2,3), the 0<x<1 f(x): 2<x<3 (a) Show that f is continuous and…

A: Given, f=x0≤x≤14-x2≤x&lt;3 S=[0,1] ∪ [2,3) (a). Since the function is linear on S, the function is…

Q: 3 Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0 < |e'(x)| < |f'(x)| for all x €…

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Q: 4. If f(x) and g(x) are continuous on [a, b], which one of the following statements is true? [f{x)–…

A: Solution:

Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

Q: 2.) Determine the set of points at which the function is continuous. а. F(x,у) — 1+x²+y² 1-x2-у? ху…

A: Here, to find the set of points at which the function is continous.

Q: Let h: R → R be a continuous function on all of R. If h(x) T , prove that h(T)<0.

A: Given that, let h:ℝ→ℝ be a continuous function on all of ℝ. Also given, hx≤0 for all x&gt;π.

Q: Let f and g be continuous funCHons such that So f(x) dx =9,S f(x) dx = 5, and S g(x)dx= -7 what is…

A: Given, ∫06fxdx=9∫36fxdx=5∫30gxdx=-7

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: 8. Given for C[0,1] the inner-product function (f,g)=[f(x)g(x)dx , find ||f|| if f(x) = x+1.

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Q: Suppose that |- f(x) is not constant. Il- f(x) is differentiable on 0, o0 II- S f(t)dt = (x +1)f(x)…

A: In the given question we have to choose a function which satisfies the given three condition in the…

Q: ( x+y if (x,y) # (0,0) Q.3 ( 3+3+4) (a) Prove that the function f(x,y) = 2x+y is not continuous (1…

A: We apply two path test for proving the discontinuity of function

Q: f : R → R be defined by f(x) = x² + x. Pro nition, that f(x) is continuous at all a E R.

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Q: Find the Wronskian for the set of functions.{x, ex, sin x, cos x}

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Q: Suppose f(x) is continuous on R such that F(x) is an antiderivative of f(x). Furthermore i Find F(1)…

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Q: For the IVP, y'=f(x,y), y=yo, x=xo, iff and df/dy are continuous discontinous defined undefined then…

A: Here we have to choose the correct option.

Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

Q: Let Y be any topological space and y: [0, 1] → Y be a continuous function so that r(0) = r(1) = xp.…

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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: Prove that there is a unique continuous function y : [0, 1] → R such that y(r²) x E (0,1]. y(x) = e"…

A: Differentiate the equation wrt x:  ddxyx=ddxex+yx22y'x=ddxex+ddxyx22y'x=ex+xy'x2

Q: 4. For a function f:D → R and a point xo in D, define A = {x in D x ≥ xo} and B: {x in D x ≤ xo).…

A: Recall definition of conituity as, Right hand limit = left hand limit = value of function at the…

Q: fhow thot te function f depined by f(XrY) = 1 (x,Y) =(1,-リ - 1) %3D x2 +y (x, y) F (I,-1) * t y is…

A: We use the sequential definition of continuity to solve this problem.

Q: 3. Determine the set of points at which the function is continuous. (a) F(x, y) = V1+x – y (b)…

A: To find the sets where the function is continous. Note that, the function is continuous on a set…

Q: 5. Let C be the set of all continuous functions f :[0,2]→ R . Define A:C→ R by A(f)= S f(x)dx . A is…

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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: 45. Let f(x, y) = mxy², where m is a constant. Find a valuc of m such f(x, y)dA = 1, where R = [0,…

A: It is given that, fx,y=mxy2, where m is a constant.We have to find the value of m such that,∫∫Rfx,y…

Q: d(x, A) = inf {|x – y| : y E A}. Show that d(x, A) = 0 if and only if x is an element of the closure…

A: As you asked for question number 7 , I solve this for you.

Q: Q4(a)- If fE R(a)on [a, b, a < x < b and F(x) = S* f(t) dt] then F(x)is continuous on [a, b].

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Q: For 1 eR, consider the boundary value dy d'y +2x x +Ay = 0, xe[1,2]| -(P.) problem dx y(1) = y(2) =…

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Q: The limit of f (x, y) = yx3 as (x, y) approaches (0,0): %3D x6+v2 O Infinity Doesn't exist O Exists…

A: Evaluate limx,y→0,0 yx3x6+y2

Q: J. Suppose that X and Y have a continuous joint distribu- tion for which the joint p.d.f. is as…

A: a)Here,fxX=∫01fx,ydy=∫01x+ydy=∫01xdy+∫01ydy=xy01+y2201=x+12…

Q: For a function h and an interval I, the oscillation of h on I is defined by w(h, I) = sup |h(æ) –…

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Q: 1. Find if the function is continuous in the given points (x, y) = (0,0) (x, y) = (0,0) x + y = 0 x…

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Q: Let Ly = y" + py' + qy. Suppose that yi and y2 are two functions such that Lyi = f(x) and Ly2 =…

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Q: 3. Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) =…

A: Given:- f: R2→R

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: 3. Evaluate the limit r(t +h) - r(t) lim h→0 h for r(t) = (t-1, sin(t), 4).

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Q: LET 2.3 + xy? if (x, y) # (0,0) f(x, y) = x² + 2y4 if (x, y) = (0,0). Show that f is not continuous…

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Q: Given F(x, y) = (2xe'6(y), e'x²(eY + 0(y)) with o E C'(R) then O (A) F is conservative only if o(y)…

A: (A ) is true.

Q: 6x2 - x²y +6y2 x2 + y2 Define f(0, 0) in a way that extends f(x, y) = to be continuous at the…

A: The given function is- fx,y=lnx2y-x Partially differentiating it w.r.t. x- ∂f∂x=2xy-1x2y-x Again…