2. A function f A → R is Lip- schitz if there exists an M> 0 so that for all x, y E A with x y we have f(x)-f(y)

Q: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

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Q: 3. Let f, g to be two continuous functions on [a, b] and f(a) ≥ g(a), f(b) ≤ g(b). Prove that Exo €…

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Q: Determine, with justification, whether the series converges absolutely, converges conditionally, or…

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Q: Indicate whether the following statement is true or false. Let f(x, y) be any c' real valued…

A: Given fx,y be the C1 real valued function defined on the rectangle R=a,b×c,d. Then we have to check…

Q: b) Give an example of a function g: R →R that is continuous at some xE R, but not differentiable at…

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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.

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Q: 4. Let f: R→ R be defined by f(x) = Prove that f is continuous at 0. Sx, TEQ, (-x, -x, xER\Q.

A: Definition:A function f(x) is continuous at x=a when,lim x→a– f(x)=lim x→a+ f(x)=f(a)

Q: .Consider the function f(7,4) = { xy x²+4y? if (x, y) # (0,0) if (x, y) = (0,0) O Is f continuous at…

A: Since the given function is of type function of several variable therefore

Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

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Q: Define f : R → R by ƒ(x) = [x]. Determine the collection of points C on which f is differentiable,…

A: The solution is below.

Q: 5. Assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). Prove there are x, y = [0, 1] with |x -…

A: We have to assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). and prove that there are x, y ∈…

Q: 15. Let f(x, y) = 22 if (x, y) ‡ (0,0), and f(0,0) = 0. Is ƒ continuous at (0,0)? x+y

A: