Let f : [a, b] → R be a monotone function. Show that the discontinuity set...

Let f : [a, b] → R be a monotone function.

Show that the discontinuity set Disc(f) is countable.

Expert Solution

Colby Messinger answered 9 months ago

4. Let a < b and f be monotone on [a, b]. Prove that f is...

4. Let a < b and f be monotone on [a, b]. Prove that f is Riemann integrable on [a, b].

Let the schema R = (A,B,C) and the set F = {A → B,C → B}...

Let the schema R = (A,B,C) and the set F = {A → B,C → B} of FDs be given. Is R in 3NF? Why or why not?

Let A ⊆ R, let f : A → R be a function, and let c...

Let A ⊆ R, let f : A → R be a function, and let c be a limit point of A. Suppose that a student copied down the following definition of the limit of f at c: “we say that limx→c f(x) = L provided that, for all ε > 0, there exists a δ ≥ 0 such that if 0 < |x − c| < δ and x ∈ A, then |f(x) − L| < ε”. What was...

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

Suppose f : R → R is measurable and g : R → R is monotone....

Suppose f : R → R is measurable and g : R → R is monotone. Prove that g ◦ f is measurable

Let a < c < b, and let f be defined on [a,b]. Show that f...

Let a < c < b, and let f be defined on [a,b]. Show that f ∈ R[a,b] if and only if f ∈ R[a, c] and f ∈ R[c, b]. Moreover, Integral a,b f = integral a,c f + integral c,b f .

Let f:(a, b) → R be a function and n∈N. Assume that f is n-times differentiable...

Let f:(a, b) → R be a function and n∈N. Assume that f is n-times differentiable and f^(n)(x) = 0 for all x∈(a,b). Show that f is a polynomial of degree at most n−1.

6. (a) let f : R → R be a function defined by f(x) = x...

6. (a) let f : R → R be a function defined by f(x) = x + 4 if x ≤ 1 ax + b if 1 < x ≤ 3 3x x 8 if x > 3 Find the values of a and b that makes f(x) continuous on R. [10 marks] (b) Find the derivative of f(x) = tann 1 1 ∞x 1 + x . [15 marks] (c) Find f 0 (x) using logarithmic differentiation, where f(x)...

5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f...

5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f is injective, but not surjective. (b) Suppose g : R\{−1} → R\{a} is a function such that g(x) = x−1, where a ∈ R. Determine x+1 a, show that g is bijective and determine its inverse function.

Suppose a function f : R → R is continuous with f(0) = 1. Show that...

Suppose a function f : R → R is continuous with f(0) = 1. Show that if there is a positive number x0 for which f(x0) = 0, then there is a smallest positive number p for which f(p) = 0. (Hint: Consider the set {x | x > 0, f(x) = 0}.)

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