Exercise 2.3.4. Let (X,dx), (Y, dy), (Z, dz) be metric spaces, and let f : XY and g Y Z be two uniformly continuous functions. Show that gof: XZ is also uniformly continuous. 36 2. Continuous functions on metric spaces Exercise 2.3.5. Let (X, dx) be a metric space, and let f: X→ R and g : X → R be uniformly continuous functions. Show that the direct sum fog: XR² defined by fg(x) := (f(x), g(x)) is uniformly continuous. Exercise 2.3.6. Show that the addition function (x, y) x + y and the sub- traction function (x, y) ⇒ x − y are uniformly continuous from R² to R, but the multiplication function (x, y) xy is not. Conclude that if f : X → R and g : X → R are uniformly continuous functions on a metric space (X,d), then f+g: XR and f-g: X → R are also uniformly continuous. Give an example to show that fg: XR need not be uniformly continuous. What is the situation for max(f, g), min(f, g), f/g, and cf for a real number c?

College Algebra
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ISBN:9781938168383
Author:Jay Abramson
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Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Exercise 2.3.4. Let (X,dx), (Y, dy), (Z, dz) be metric spaces, and let f :
XY and g Y Z be two uniformly continuous functions. Show that
gof: XZ is also uniformly continuous.
36
2. Continuous functions on metric spaces
Exercise 2.3.5. Let (X, dx) be a metric space, and let f: X→ R and g : X →
R be uniformly continuous functions. Show that the direct sum fog: XR²
defined by fg(x) := (f(x), g(x)) is uniformly continuous.
Exercise 2.3.6. Show that the addition function (x, y) x + y and the sub-
traction function (x, y) ⇒ x − y are uniformly continuous from R² to R, but
the multiplication function (x, y) xy is not. Conclude that if f : X → R
and g : X → R are uniformly continuous functions on a metric space (X,d),
then f+g: XR and f-g: X → R are also uniformly continuous. Give an
example to show that fg: XR need not be uniformly continuous. What is
the situation for max(f, g), min(f, g), f/g, and cf for a real number c?
Transcribed Image Text:Exercise 2.3.4. Let (X,dx), (Y, dy), (Z, dz) be metric spaces, and let f : XY and g Y Z be two uniformly continuous functions. Show that gof: XZ is also uniformly continuous. 36 2. Continuous functions on metric spaces Exercise 2.3.5. Let (X, dx) be a metric space, and let f: X→ R and g : X → R be uniformly continuous functions. Show that the direct sum fog: XR² defined by fg(x) := (f(x), g(x)) is uniformly continuous. Exercise 2.3.6. Show that the addition function (x, y) x + y and the sub- traction function (x, y) ⇒ x − y are uniformly continuous from R² to R, but the multiplication function (x, y) xy is not. Conclude that if f : X → R and g : X → R are uniformly continuous functions on a metric space (X,d), then f+g: XR and f-g: X → R are also uniformly continuous. Give an example to show that fg: XR need not be uniformly continuous. What is the situation for max(f, g), min(f, g), f/g, and cf for a real number c?
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