Let f:R→R satisfy f(x+y) =f(x) +f(y), ∀x,y∈R. If limx→0f(x) exists, prove that f is continuous on R

Q: Let f(x) = sin(sin(sin.x)) for for all x in R (f(x) x-0 g (x), and g(x) = (sinx), VxER, then lim is…

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Q: Let h: R → R be a continuous function on all of R. If h(x) Tt, prove that h(t) 50.

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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: 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: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

A: Solution is given below:

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: Suppose that f :R -R is differentiable at z = 0. (a) Prove that (t) lim,o E = f'(0) (b) Give an…

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Q: Show that the limit of the function f(x,y)= (0,0). (x+y)² x² + y² is indeterminate at

A: Limit of the function

Q: x² – 6x, { 21? — 3и — 4, 1 4.

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Q: Show that the function f(x, y) x4 + y? has no limit as (x, y) approaches (0,0).

A: Note: Since we can solve at most one problem at a time, we have solved the first one. Kindly…

Q: Prove that f(z)=|z| is nowhere differentiable in C but is continuous everywhere.

A: Given function is fz=z. We have to prove that the function fz is nowhere differentiable in C but is…

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: Compute f'(a) in two ways, using Eq.(1) and Eq.(2). f(x) — 6х3, а %— 2 lim h→0 f(a + h) – f(a) h…

A: Given that f(x) = 6x3  , a =2we have to find f'(a) = limh→0 f(a+h)-f(a)hfirst find f(a+h)…

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

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Q: If f(z) is continuous at z, then it must be differenttiable at z, ylgn O İhi O

A: If a function f(z) is continuous at z0 , then it may or may not be differentiable at z0. Hence, The…

Q: f(z) = Im z is continuous

A: Let us use the convergence criterion to prove that Im z is continuous. Let us assume that  znn=1∞ be…

Q: Let y = f(x) be a continuous function. If ƒ has a local maximum or minimum value at x = a, then the…

A: Let y=f(x) be a continuous function. If f has a local maximum or minimum at x=a,then the limit…

Q: Define f(0,0) in a way that extends f to be continuous at the origin. 2.2 11x - xy + 11y 2 f(x,y) =…

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)Df(x)+fy) vx, yER. Show that…

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Q: Let f be a continuous function with domain R. Compute the limit (z- t) f(t) lim dt.

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Q: Prove that if f (x) is continuous at x = a and g(y) is continuous at y = b, then F(x,y) = f(x)g(y)…

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Q: Let f(x.y) be a function defined in R2. If lim,-of(x, kx) = 4 for all real numbers k and f(x.y) is…

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Q: If f is continuous everywhere except at x = 2, then f(x +1) dx is not integrable. True False

A: To check whether the given statement is True or False."If f is continuous everywhere except at x=2,…

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

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Q: Prove that if f, g :R → R are continuous at x E R, then ƒ+gis continuous at x E R as well.

A: By using the definition of continuous function this above statement can be easily proved.

Q: Consider the line f(x) = mx + b, where m ≠ 0. Use the ε- definition of limit to prove that limx c…

A: We have given function fx=mx+b, where m≠0. We have to prove that limx→cfx=mc+b using ∈,δ definition.…

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: Suppose that f : R -R is differentiable at z = 0. (a) Prove that (t) lim,-o EA = f'(0) (b) Give an…

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Q: use the formal definition of limit to prove that thefunction is continuous at c. F(x) = √,9 - x, c =…

A: Given: Fx=9-x, c=5 we know that any functionfx is continuous at given point x=c if,…

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: Show that the limit of the function f(x,y) = at (0,0) does not exist. (x+y)² x² + y²

A: Limit of a Function: Like a single-variable function, a multi-variable function can also have limit.…

Q: Suppose f : R+ R is continuous and f(0) = 0. Consider the limit yf(x) lim (x,y) (0,0) y² + f (x)2*…

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Q: 7. Let f(2) = f(x, y) = + i , = 0. when z +0 and let f(0) %3D 5xº+y² (a) Show that lim f(z) = f(0) =…

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Q: Let the function f: R R be differentiable at x = 0. Prove that lim %3D = 0.

A: Given the function f:R→R be differentiable function at x=0. We have to prove that limx→0fx2−f0x=0.…

Q: Let f, g : D → R be continuous at c E D. Prove that fg is continuous at c.

A: Let ε&lt;1 and M= maxf(a), g(a), 1, then there exists δ&gt;0 such that f(x)-f(a)&lt;ε3M (since f is…

Q: Prove that if S(c +h) – f(c) lim h exists, then f is continuous at c.

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

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Q: (b) Let h(r, y) be a function defined by h(r, y) = h(0, 0) = 0 3r2 + y? (i) Prove that (ii) Using…

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Q: Find the limit of f(x) as x approaches 2, where

A: fx=1,  x≠20,  x=2

Q: Let f : R → R be defined by (b) f(x) = *. Use the e – 6 definition to prove that f is continuous on…

A: We have to use epsilon-delta definition of continuity and limit

Q: Let fn f, where if 0<<! Sn(x) ={ S 2-a if 0<x<1 and f(x) if a = 0 Then lim sup fn (x) – ƒ (x)| =

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Q: Let A := (0, 1] and let ƒ : A → R be defined by f(x) = !. Prove that ƒ is continuous on A.

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Q: Prove that the limit of xy[(x^2 - y^2)/(x^2 + y^2)] is zero as (x, y) goes to (0,0)

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Q: Consider the function { that is defined on R as: f(x)=-x² +4x-3. Designate by (C) its representative…

A: The given function is, fx=-x2+4x-3 The orthonormal system is, O; i→,j→ To evaluate limx→-∞fx  and…

Q: . Is f(x)= Vx + 1 continuous at x == -1? Yes, because all functions are continuous No, because the…

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Q: Let f : AC X → Y be a continuous function between metric spaces. If A is open then so is f(A).

A: Continuous Function: Let (X, dX) and (Y, dY ) be metric spaces. A function f : X → Y is continuous…

Q: Let A := (0,1] and let f : A → R be defined by f(x) = . Prove that f is continuous on A.

A: Consider the provided question,

Q: If X is a continuous RV, prove that E(X) = [1- F(x)] dx – F(x) dx. %3D

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