"Prove that the function ƒ : R → R with ƒ(x) = 1 is uniformly continuous." "A function f : A → R is called uniformly continuous if for every € > 0 there exists a § > 0 such that for all x, yin A, |x − y| < d implies |ƒ(x) − ƒ(y)| < e."

"Prove that the function ƒ : R → R with ƒ(x) = 1 is uniformly continuous." "A function f : A → R is called uniformly continuous if for every € > 0 there exists a § > 0 such that for all x, yin A, |x − y| < d implies |ƒ(x) − ƒ(y)| < e."

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

See similar textbooks

Similar questions

Show that if g: R R is a function which is differentiable and with g' bounded (say |g'(x) 0 is one-to-one for all sufficiently small .
Let f be a real-valued continuous and differentiable function. Let function g be defined by g(x) = f(|x| + 2). A student presents the following proof to show that there exists a real number c € (-1, 1) such that g/(c) = 0. (1) Since f is a continuous function, so is g over the interval [-1, 1]. (II) Since f is differentiable, so is g over the interval (-1, 1). (III) It is evident from the definition of g that g(−1) = g(1). (IV) If the above conditions hold, then by Rolle's theorem, there exists dg(x) c = (-1, 1) such that gl (c) dx |x=c Which statement about this proof is correct? = 0. Step (1) does not hold, and hence Rolle's theorem does not apply. Step (II) does not hold, and hence Rolle's theorem does not apply. Step (III) does not hold, and hence Rolle's theorem does not apply. Step (IV) does not hold, and hence the conclusion is false. The proof is completely correct, and the conclusion holds.
L. Let f(z) be an entire function. (a) If f'(z)| ≤ z for all z € C, prove that there exists a, 3 € C such that f(z) = a + 3z² for all z EC and 3| ≤ 1. (b) If f(2)= f(z + 1) = f(z+i) for all z € C, prove that f(z) is constant. (c) If |f(z)| →∞ as z →→→∞, prove that f(z) is a polynomial function.
Denote the range of a function f, as z varies over a domain D, by f(D). The function f is called bounded away from zero on the domain D if zero does not belong to f(D) and does not belong to the boundary of f(D). Determine whether the function f(z) = e is bounded away from zero on (b) D = {T/4 < Imz < T/2}.
2. Let f : R → R be a continuous function. Suppose that (f o f) (x) = x for every x E R. (a) Prove that there exists a point c ER such that f (c) Hint: Consider the function g (x) = f (x) – x for every x € R. = C. (b) Give an example for a function satisfying the above requirements, and such that f id, -id , where id is the identity function.
Define f : R → R by ƒ(x) = x². Compute the following, or determine if they do not exist. min ((-1,3)) (b) inf ((-1,3)) (c) min (f((-1,3))) (d) inf (ƒ((−1,3)))
only (c) and don't use AI
Suppose f: (0, ∞) → (0, ∞) and g: (0, ∞) → (0, ∞) and h: (0, ∞) → (0, ∞) are continuous functions such that h(x) = f(x)g(x) for every positive real number . True or false: If f and h are differentiable, then g must be differentiable too. True False
Show that the functions(a) f(x)=1/(1+x2),x∈R, (b) g(x)=x/(1+x2),x∈R, are uniformly continuous, while the functions(c) f(x)=x2,x∈R, (d) g(x)=√∣x∣,x∈R, are not Through graph
Suppose f: (0, ∞) → (0, ∞) is a continuous function, and g: (0, ∞) → (0,00) is defined by the property that g(x) = (f(x))² for every positive real number . True or false: If g is differentiable, then f must be differentiable too. True False