14. Suppose f : R → R is continuous. Let a, < c < bn and an, bn → c as n → 0. Show that 1 b. - 0. f(x) dx → f(c) as n → ∞. an

Q: 45. Let g(x) = [x]. Find a) g-¹({0}). c) g-¹({x|0<x< 1}). b) g-¹({-1, 0, 1)).

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Q: Suppose f:R→IR is a diffexetiuble function which satısfies flo) = 0 . We wort to shouw f': IR → IR…

A: We have to do implicit differentiation.

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: 18. Let f(x) be continuous on [1, 3] and f (1) = 4, f(3) = 20. Show using the intermediate value…

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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: (x. v) = (0, 0)

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Q: 7. Suppose f is a continuous function on (a, b) with b> a. Stat f (recall this theorem was taught in…

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Q: Q4) let f be non-negative continuous function of [a, o]& k> 0 is constant. IF t f(t) a holds, then…

A: Given f is a non-negative continuous function on [a,∞) and k&gt;0 is a constant. Suppose,…

Q: 5. Let f : R -→ R be defined by f(x) = 2x +9. Find a formula for f-1.

A: We need to find the inverse function of the real valued function f(x)=2x+9

Q: Let f(r) = In r Find and simplify f"(x).

A: Please refer the attached image for complete solution. THANK YOU.

Q: 7. Suppose that f: R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the…

A: Recall the definition of continuity around c .And then choose an ∈&gt;0 to get a δ&gt;0…

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

A:

Q: 4. (a) Let f: X→Y, g: Y→ Z, and h: Y→ Z be functions. Prove or disprove: If gof=hof, then g = h. (b)…

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Q: 4. Let f and g be functions defined on an open interval I containing the point c. Prove that if f…

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Q: 6. Let f: Q R be defined by f(x) Show that f is continuous. 0, :{ 1, = if x € (-∞, √2) nQ if x €…

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Q: 84. Let f be a continuous function, c a real number. Show that cb+c (a). | f(x – c) dx = | f(x)dx,…

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Q: 6. Let S = {-1,0, 2, 4, 7}. Find f(S) (image of S under f) where f is from R to R for (a) f(a) = 1…

A: The

Q: 6. Let f be a function such that f(2) = 3 and f(6) = 19. Which of the following conditions…

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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: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

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Q: each

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Q: 4. If continuous function f:[0,1] U (2,3) → R is one-to-one, is f also continuous? 1

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Q: If that f is continuous on a sxsb. Let F(x) be any of f(x). Then, f(x)dx = F(b) - F(a). This theorem…

A: Recall: Mean value theorem: If f(x) is continuous over an interval [a,b], then there is atleast one…

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: Which of the following functions is continuous on the interval (-0, +)? …* f(x) = Vx² – 1 (A f(x) =…

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Q: Define h as x 2 Determine whether the function h is continuous h(x) = {3, x2 or discontinuous at x…

A: NOTE: Refresh your page if you can't see any equations. . if the function h(x) is continuous at x=2…

Q: 3. Define f(x) so that S(2) = {ap(). f(x) = 0, |피 1 (a) Explain why the function f(x) must be…

A: As per our guidelines we can answer only 3 subparts. please ask others seperately.

Q: A fixed point of a function f is somer where f(x) = x. Suppose that f: [a, b] → R is differentiable…

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Q: 3. Let there be some a, b R such that f(a)ƒ'(a) = f(b)ƒ'(b) + 1. Show that f f(x) f"(x)dx ≤ 1.

A: This is a problem related to integration. Based on the general formula and concept we will prove the…

Q: Let n > 2 be a natural number. Prove that there is no continuous function q : C → C such that…

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Q: 15. Let f, g : R → R be continuous at a point c, and let h(x) = sup (f (x). 8(x)} for x e R. Show…

A: A function is one-one means if the value of the domain are equal then the value of the range set…