Advanced Calculus 1 Problem 1 If the function f : D → R is uniformly continuous...

Advanced Calculus 1

Problem 1 If the function f : D → R is uniformly continuous and α is any number, show that the function αf : D → R also is uniformly continuous.

Problem2 Provethatiff:D→Randg:D→Rareuniformlycontinuousthensois the sum f + g : D → R.

Problem 3 Define f (x) = 2x + 1 for all x ∈ R. Prove that f is uniformly continuous.

Problem 4 Define f (x) = x3 + 1 for all x ∈ R. Prove that f is not uniformly continuous.

Expert Solution

Colby Messinger answered 4 months ago

1) There is a continuous function from [1, 4] to R that is not uniformly continuous....

1) There is a continuous function from [1, 4] to R that is not uniformly continuous. True or False and justify your answer. 2) Suppose f : D : →R be a function that satisfies the following condition: There exists a real number C ≥ 0 such that |f(u) − f(v)| ≤ C|u − v| for all u, v ∈ D. Prove that f is uniformly continuous on D Definition of uniformly continuous: A function f: D→R is called uniformly...

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}.)

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.

a) Let S ⊂ R, assuming that f : S → R is a continuous function,...

a) Let S ⊂ R, assuming that f : S → R is a continuous function, if the image set {f(x); x ∈ S} is unbounded prove that S is unbounded. b) Let f : [0, 100] → R be a continuous function such that f(0) = f(2), f(98) = f(100) and the function g(x) := f(x+ 1)−f(x) is equal to zero in at most two points of the interval [0, 100]. Prove that (f(50) − f(49))(f(25) − f(24)) >...

1)Using R, construct one plot of the density function for a uniformly distributed continuous random variable...

1)Using R, construct one plot of the density function for a uniformly distributed continuous random variable defined between 0 and 6. Make sure you include a main title, label both axis correctly and include a legend in your figure. The main title needs to include your last name, also include your R code with your answer. 2) Using R, construct one plot of the density function for a Chi-square distributed random variable. The plots should contain six lines corresponding to...

Give an example of a continuous function that is not uniformly continuous. Be specific about the...

Give an example of a continuous function that is not uniformly continuous. Be specific about the domain of the function.

6.3.8. Problem. Let f : A → B be a continuous bijection between subsets of R....

6.3.8. Problem. Let f : A → B be a continuous bijection between subsets of R. (a) Show by example that f need not be a homeomorphism. (b) Show that if A is compact, then f must be a homeomorphism. 6.3.9. Problem. Find in Q a set which is both relatively closed and bounded but which is not compact.

1) The graph of the derivative f ' of a continuous function f is shown. (Assume...

1) The graph of the derivative f ' of a continuous function f is shown. (Assume the function f is defined only for 0 < x < ∞.) (a) On what interval(s) is f increasing? (2a) On what interval(s) is f decreasing? (b) At what value(s) of x does f have a local maximum? (2b) At what value(s) of x does f have a local minimum? x=? (c) On what interval(s) is f concave upward? (2c) On what interval(s) is...

(Advanced Calculus and Real Analysis) - Cantor set, Cantor function * (a) Define the Cantor function....

(Advanced Calculus and Real Analysis) - Cantor set, Cantor function * (a) Define the Cantor function. (b) Prove that the Cantor function is non-decreasing.

Let φ : R −→ R be a continuous function and X a subset of R...

Let φ : R −→ R be a continuous function and X a subset of R with closure X' such that φ(x) = 1 for any x ∈ X. Prove that φ(x) = 1 for all x ∈ X.'

Question