Exercise 4: (Hard) Give an example of a function f with domain [0, 1] with infinitely many points of discon- tinuity such that f is integrable from 0 to Hint: f can be discontinuous on the set {n-1, n e N}. Remark: There is no need to formally prove that f is integrable, a good explanation is fine.

Q: interval [a,b] where it is NOT integrable. f(x)=           ,   x∈

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Q: Prove that if ƒ and g are Riemann integrable on [a, b] then fg is also Riemann integrable on [a, b].…

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Q: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

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Q: Define f : Rm \ {0} → R by ƒ(x) = T: Prove that f is uniformly continuous on R" \ B1(0), but not…

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Q: Define f: [0, 1] →→ R by the property that 1, if is rational, f(x) = a 0, if ar is irrational. True…

A: f:0, 1→ℝfx=1, if x is rational,0, if x is irrational.

Q: Let f(x) = 9x2, x in QQ = 0, x not in QQ (a) Prove that f is continuous at exactly one point…

A: f(x)=9x2x∈Q0x∉Q

Q: Consider the function f : defined by                    f(x) = 0 if x is irrational and f(x)=1 if x…

A: The objective of the question is to determine whether the given function f, which is defined…

Q: kx +5 for x ≤ 2 Let fx = = x-1 for x > 2 Find the value of k for which f(x) is continuous at x=2.…

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Q: (i) Using scaling and translation invariance only, find a, b,c such that for some Rie- mann…

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Q: Let f(z) be holomorphic on C (entire function). (a) Show that for any a, b e C, there exists R> 0…

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Q: Is the proof correct? Either state that it is, or circle the first error and explain what is…

A: The given proof is correct.

Q: Exercise 4: Suppose f is differentiable on (2,6) and continuous on [2,6. Suppose further that 1 < f'…

A: Given:f is differentiable on (2, 6) and continuous on [2, 6].

Q: Consider the function g :ℕ→ℝ  where  g(n) = 1 if n is odd and g(n) = 2 if n is even. Is g continuous…

A: Given function is g :ℕ→ℝ  where  g(n) = 1 if n is odd and g(n) = 2 if n is even.

Q: 1 4. Let f be analytic on C. Prove that - 2n J0 \f(ret®)|d0 is bounded for 0 <r<∞ if and only if f…

A: To Prove :- 12π∫02πfreiθdθ is bounded for all 0≤r&lt;∞ if and only if f is bounded on ℂ. Given:- let…

Q: f is Riemann integrable iff f is bounded and f has finite set of discontinuous points.

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Q: Consider the function f: RR defined by f(x) = 0 if x is irrational and f(x) = 1 if x is rational Is…

A: Note: f is said to be Riemann integrable on [a, b] if To Prove Dirichlet's function which is defined…

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

A: Given function is fx=x−1x+1. We have to show that fx=x−1x+1 is uniformly continuous on 0,∞. Let us…

Q: 3) Show that the functions max: R" R. (₁,...,n) max(₁,....n) and min: R" → R. (₁.) → min(x₁,....)…

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Q: All continuous f:[0, e function a>O impshding a2 S x*f(x*) dx = S xf(xIdx a) show +hat

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Q: Prove that for any Riemann integrable function f on the interval [a, b] there is c such that f = [°…

A: let f be Riemann integral an [a, b] then clearly f is bounded.