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?
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 (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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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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