Topology question: Show that a function f : ℝ → ℝ is continuous in the ε...

Topology question:

Show that a function f : ℝ → ℝ is continuous in the ε − δ definition of continuity if and only if, for every x ∈ ℝ and every open set U containing f(x), there exists a neighborhood V of x such that f(V) ⊂ U.

Expert Solution

Colby Messinger answered 2 years ago

Show that "f(x) = x^3" is continuous on all of ℝ

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}.)

Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ...

Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ and ?(?) = ? 2 if ? ∈ ℝ ∖ ℚ. Find all points at which ? is continuous

TOPOLOGY Let f : X → Y be a function. Prove that f is one-to-one and...

TOPOLOGY Let f : X → Y be a function. Prove that f is one-to-one and onto if and only if f[A^c] = (f[A])^c for every subset A of X. (prove both directions)

Define a function ?∶ ℝ→ℝ by ?(?)={?+1,[?] ?? ??? ?−1,[?]?? ???? where [x] is the integer...

Define a function ?∶ ℝ→ℝ by ?(?)={?+1,[?] ?? ??? ?−1,[?]?? ???? where [x] is the integer part function. Is ? injective? (b) Verify if the following function is bijective. If it is bijective, determine its inverse. ?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)

Topology question: Prove that a bijection f : X → Y is a homeomorphism if and...

Topology question: Prove that a bijection f : X → Y is a homeomorphism if and only if f and f−1 map closed sets to closed sets.

Part A. If a function f has a derivative at x not. then f is continuous...

Part A. If a function f has a derivative at x not. then f is continuous at x not. (How do you get the converse?) Part B. 1) There exist an arbitrary x for all y (x+y=0). Is false but why? 2) For all x there exists a unique y (y=x^2) Is true but why? 3) For all x there exist a unique y (y^2=x) Is true but why?

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.

1) Show the absolute value function f(x) = |x| is continuous at every point. 2) Suppose...

1) Show the absolute value function f(x) = |x| is continuous at every point. 2) Suppose A and B are sets then define the cartesian product A * B Please answer both the questions.

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f...

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f is non-decreasing on [a,b] if and only if f′(x) ≥ 0 for all x ∈ (a,b), while if f is non-increasing on [a,b] if and only if f′(x) ≤ 0 for all x ∈ (a, b). can you please prove this theorem? thank you!

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