Let I:= [a, b] and let f: I -> R be a continuous wetion such that f(x) >0 for each x iN I. Prove that here exists a Number of >0 such that f(x) > α for all XEI.

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Q: A continuous function ƒ : [0, 1] → R satisfies ƒ(0) f(1). Show that for each integer n ≥ 1 there…

A: Let fix δ=1n, n∈ℕ Consider the continuous function gnx:0,1-δ→ℝ; gnx:=fx+1n-fx and suppose that gnx≠0…

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)3Df(x)+f) x, yER. Show that f(x)3Dcx…

A: Since you have posted a multiple questions. So,  we will solve only first question  for you. To get…

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Q: = √x + 2,-2 ≤ x ≤2 ax²-bx, x+1 x-2 ) 2 3

A: Topic:- calculus

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Q: 15) Prove that if ffx) is continuous at x=0 then so is V f(x) provided that fix) zo for all x in the…

A: Step 1: Step 2: Step 3: Step 4:

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Q: Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

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Q: Let A := (0, 1] and let ƒ : A → R be defined by f(x) = !. Prove that ƒ is continuous on A.

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Q: 4. Let A := (0, 1] and let f : A → R be defined by f(x) = !. Prove that f is continuous on A.

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