3. Let f: [a, b] → R. Suppose f is continuous at c € [a, b] and f(c) > 0. Prove that there exists >0 such that f(x) > 0 for any ze [a, b] with |x-c|

Q: 2. Show that if f is continuous on [a, b] with f(x) > 0 for all a ≤ x ≤ b, then 1/f is bounded on…

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Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

Q: 1. Show that the following functions f : R R are continuous. You should argue carefully from the…

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Q: Suppose that the function f : R → R is continuous for x ≥ 0, differentiable for x > 0, and satisfies…

A: Mean Value Theorem Suppose  f ( x )  is a function that satisfies both of the following conditions.…

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: 5. Let f : R → R. Prove that f is continuous on R + f-(H) is a closed set whenever H is a closed…

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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: Let f be a twice-differentiable function defined on an open interval I. Assume there are three…

A: f be a twice differentiable function on an open interval I, and there are three points in the graph…

Q: 3. Assume f: ZR and prove that f is uniformly continuous.

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Q: Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f : (a,…

A: Given:f:a, b→ℝ is an unbounded function. To show : f' is also unbounded On the contrary,  assume…

Q: Suppose f : [a, b] → R is is continuous and such that fg = 0 for all continuous functions g : [a, b]…

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Q: Consider the function f : N → N defined by n? if n is odd f = 2n + 1 if n is even A. Find f(A) if A…

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Q: For real numbers m and b with m + 0, show that the map f: R-→ R defined by f(x) = mx + b is…

A: Given m and b be the real number with m≠0 and f:R→R defined by fx=mx+b. We have to show that fx is…

Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

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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:

Q: 3. Let f be entire. Suppose that there exist a EC and > 0 such that f(z) & Da, e) for all z E C.…

A: Since f is entire, it is holomorphic on the entire complex plane. Now, we know that f has a root in…

Q: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

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