Suppose that f is continuous at 0, and that f satisfies f(x + y) = f(x)+ f(y) for all real numbers r, y. Prove that f is continuous at every point.

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Q: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

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Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

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Q: If f is continuous at a, then f is differentiable at a. O True O False

A: Answer: False.

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

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Q: Let Show f is defined for all x &gt; 0. Is f continuous on (0,∞)? How aboutdifferentiable?

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Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

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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: Suppose that f (x) = x·g(x) for some function g that is continuous at 0. Prove that f is…

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Q: Show by definition that f(x) = |x| : R → R is continuous on R. 2.

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Q: Show that x is continuous everywhere. r(x)-x-0 XX<0

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Q: Find all values of c such that f is continuous on (-o, 0). { 7- r, /(r) – (smaller value) (larger…

A: f(x)=7-x2,x≤cx,x&gt;c

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: f is continuous at a if and only if |f| is continuous at a.

A: False

Q: If f : R+ → R is a function such that f(x + y) = f(x)f(y) for all x,y ∈ R+. Prove that f is…

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Q: Find all values of c such that f is continuous on (-0, 0). 1- x2, f(z) = { – x², x. C = (smaller…

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Q: Let f:X→R be a Lipschitz continuous function defined on X. Show that f is uniformly continuous on X.

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Q: ずx 1

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Q: Determine the domain of the function and prove that it is continuous on its domain using Theorems…

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Q: Let f : R → R with f(x) = 3x² + 2. Prove that f is differentiable at the point –1 %3D

A: f(x) =3x2+2 f'(x) =limh→0f(-1+h) -f(-1)h        =limh→03(-1+h)2+2-3-2h        =limh→03+3h2-6h -3h…

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Q: Prove that if f : V → W is differentiable at a point a ∈ V it must be continuous there.

A: I have proved limiting value is equal to functional value

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Q: Show that x| is continuous everywhere. r(x)- x- 0 xX <0

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Q: 크) Prove that va +b 0. Prove that f is uniformly continuous on [0, 0).

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Q: Define f(0,0) in a way that extends f to be continuous at the origin. 6x² -x²y² + 6y² x² +y? f(x,y)…

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Q: Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

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Q: if x = 0 f (x) = %3D x* sin(-) otherwise

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Q: 5.Prove that: if f is continuous at x = 0, then xf (x) is differentiable at x = 0.

A: We want to prove the given result.

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A: Prove the given statement

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Q: Prove that the function f(x)=sinπ|x| is continuous at x = 0 but not differntiable at the same point.

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Q: Suppose that f is continuous on [a.b] that f(x) 20 for all xE[a,b] and that f =0 1. Prove that f(x)…

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

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,

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Q: Assume that f is continuous on [a, b] and differentiable on (a, b). Prove that if f' > 0 for all a <…

A: Assume that f is continuous on [a,b] and differentiable on (a,b). Given that  f'&gt;0 for all…

Q: Suppose that a function y = f(x) is continuous on I = [a,b] and if the set f(I) = [f(a),f(b)], then…

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Q: f (x) = %3D x² +1

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