Let f : [0, 1] → R be a function with Darboux’s property such that for any y ∈ R, the set  f ^(-1)({y}) is closed. Prove that f  is continuous.

Q: Suppose f : A → B is a function, X ⊂ A and Y ⊂ B. What does it mean to say: (a) z is in the range of…

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Q: Suppose  f ( 200 , 300 ) = 1000and ∂ f ∂ x ( 200 , 300 ) = 12 and ∂ f ∂ y ( 200 , 300 ) = 10.…

A: Given, f(200,300)=1000 ∂f∂x(200,300)=12∂f∂y(200,300)=10

Q: A continuous function defined on an open interval (a,b) must have a minimum value. True False

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Q: Suppose f is a function defined and continuous on the open interval (a, b). Suppose g is a function…

A: Given that: f is a function defined and continuous on the open interval (a, b).g is a function…

Q: Let f be defined in an open interval that contains a=3 and assume that all x,…

A: f be defined in an open interval that contains a=3 and assume that all x,2x+6≤f(x)≤x2-4x+15

Q: Prove there does not exist a continuous function which maps the closed interval [0, 1] onto the open…

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Q: Sketch a graph of a function ƒ with the following propertie. ƒ' < 0 and ƒ" < 0, for x < 3ƒ'…

A: Since f’ < 0, this means the function is decreasing for all values of x. Since f’’ < 0 for x…

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: Let f:[0,1]→R, be the function defined by, f(x)=|x-1/2|. Show that f is continuous and conclude that…

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Q: What is an example of a function f : [2, 3] -->R which is not continuous at 1 but which is…

A: Here's an example of a function that is not continuous at 1 but is integrable on the interval [2,…

Q: sider the function f (t) = –2t³ – 6t² + 8 on the interval [-1, 2]. of values of c for which such…

A: f'(c) = f(b) -f(a) /b-a

Q: Let f. ROR be a Continuous function with f(x) ≤ 10 for = all XER. Explain why f is not homeomorphism

A: A homeomorphism is a continuous bijection with continuous inverse.

Q: Let f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not…

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Q: If m, n ∈ N, then (f^m)^n = f^(mn).

A: To prove two functions are equal, we need to show that both the functions have same function values…

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: