Let f: [1,1]→ R be a continuous function. Prove the following statements: (a) If there is c = [-1, 1] such that f(c)f(−c) < 0, then there is de R such that f(d) = 0. (b) If ƒ([−1, 1]) = (-1, 1), then f is not continuous.

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Q: The function f is continuous on [0, 4]. f(0)= -2 and f(4) = 5. Must there exist a number C between 0…

A: Given,         The function f is continuous on 0,4. f0=-2 and f4= 5.         There exist a number c…

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

Q: 2. Prove: Let ƒ : R → R be a continuous function satisfying f(x) = 0 for all rational numbers x.…

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Q: If ƒ is differentiable at a, must ƒ be continuous at a?

A: we know that for a given function if it is differentiable at given point then it must be continuous…

Q: Let f(x) = |x|+|x−2], x € R. Find points where f is not differentiable.

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Q: the following statement. 'If yes, explain why. If no, provide a counterexample. For every real…

A: If the function is injective , then the above statement is true.

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Q: Consider the function f: R → R defined by f(x) = x³ – 3x + 2. Then (A) ƒ is increasing on the…

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Q: f(x) = {*- Įx² +px + 2,x # 3 3, x = 3

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Q: c. Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one…

A: We can explain this statement by 2 explainations given below

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Q: Use the e-6 definition of continuity to prove that the following functions are continuous at the…

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Q: verify 'If yes, explain why. If no, provide a counterexample. the following statement. For every…

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Q: 4. Let f:R→ R be a function defined by f(x)=x for every xeX . Show that f is a continuous.

A: Let A ⊆ R , let f:A → R and let c ∈ A. We say that f is continuous at c if, given any number ε&gt;0,…