1. Use the e-ổ definition to prove that f(x) = x² is uniformly continuous on [a, 이.

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 f: [-1,0] → R and g: [0, 1] → R are continuous and f(0) - g(0). Define h: [-1,1] → R by…

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Q: Assume that h: [a, b²] → R is a continuous function and let G : [a, b] → R denote the following…

A: Answer Explanation:

Q: Suppose f is defined on [a, b] and g is defined on [b, c] with f(b) g(b). Then define { f(x) if a <…

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Q: 3. Let f, g : [a, b] → [c, d] be cotinuous functions. Suppose that f is surjective. Prove that there…

A: We will use of intermediate value theorem to proof this

Q: Exercise 4: Suppose f is differentiable on (2,6) and continuous on [2,6. Suppose further that 1 < f'…

A: Given:f is differentiable on (2, 6) and continuous on [2, 6].

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(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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Q: Show that a continuous function f : [0, 1] → Z is a constant function.

A: Consider a function .We need to show that a continuous function  is a constant function.We know…

Q: (b) Let f: [a, b] → R be a continuous function which satisfy f(a) 0. Let W C[a, b] defined by W:=…

A: (b) Given information: f :a,b→ℝ is a continuous function which satisfy f(a) &lt;0 and f(b) &gt;0.…

Q: 31. Assume that f is continuous and 2x S f ₁ f(t) d² = 4 + x²" dt (a) Determine f(0). (b) Find the…

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Q: 7. Suppose f is a continuous function on [a, b] and Sf(x)g(x) dx =0 for every continuous function g.…

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Q: 6. Let f be a continuous function on [0, 2] whose second derivative exists on (0, 2). Assume f(0) =…

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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: Prove that f: R→ R is continuous. 2. Suppose that the function f : [0, 1] → R is continuous and that…

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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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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.