Let f be a function with domain the reals and range the reals. Assume that f...

Let f be a function with domain the reals and range the reals. Assume that f has a local minimum at each point x in its domain. (This means that, for each x ∈ R, there is an E = Ex > 0 such that, whenever | x−t |< E then f(x) ≤ f(t).) Do not assume that f is differentiable, or continuous, or anything nice like that. Prove that the image of f is countable. (Hint: When I solved this problem as a student my solution was ten pages long; however, there is a one-line solution due to Michael Spivak.)

Expert Solution

Colby Messinger answered 2 years ago

• The range of f is the domain of g and the domain of f is...

• The range of f is the domain of g and the domain of f is the range of g. • f(a) = b if and only if g(b) = a. • (a, b) is on the graph of f if and only if (b, a) is on the graph of g. The function f(x) = x3 + 7x + 5 is one-to-one. Since finding a formula for its inverse is beyond the scope of this textbook, use Properties of...

The deﬁnition of a (well-deﬁned) function f : X → Y . Meaning of domain, range,...

The deﬁnition of a (well-deﬁned) function f : X → Y . Meaning of domain, range, and co-domain

Graph the function f(x) = 3.5(2)x. State the domain and range and give the y-intercept.

Let f be a continuous function on the closed interval [0,1] with a range also contained...

Let f be a continuous function on the closed interval [0,1] with a range also contained in [0,1]. Prove that f that there exists an x in [0,1] such that f(x)=x. Is the same explanation still valid if f is not continuous?

Domain and range

What is the domain and range of y=1/2x²+4?

why are the domain and range of square root function restricted to [0,∞) ?

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

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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...

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