16.D. Let f and g be continuous functions on Rº to R and let h, k be defined for x in Rº by h(x) = sup {f(x), g(x)}, k(x) inf {f(x), g(x)}. Show that h and k are continuous on R. [Hint: use the relations sup {a, b} = 글 {a tb+ la-히}, inf {a, b} =D {a +b-la-b1]
16.D. Let f and g be continuous functions on Rº to R and let h, k be defined for x in Rº by h(x) = sup {f(x), g(x)}, k(x) inf {f(x), g(x)}. Show that h and k are continuous on R. [Hint: use the relations sup {a, b} = 글 {a tb+ la-히}, inf {a, b} =D {a +b-la-b1]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Help with exercise 16.D.
Please use the corollary, and demonstrate the exercise in detail (Show each statement that is given)
![16.2 COROLLARY. Let f be defined on all of Rº and with range in Rº.
Then the following statements are equivalent:
(a) f is continuous on RP.
(b) If G is open in Rº, then f-G) is open in Rº.
(c) If H is closed in Rº, then (H) is closed in RP.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6de8529-bcc7-4d6a-82fd-d13957834b62%2F34cfdc06-fc00-4702-8c25-6e307991e1ff%2Ffyxzrfe_processed.png&w=3840&q=75)
Transcribed Image Text:16.2 COROLLARY. Let f be defined on all of Rº and with range in Rº.
Then the following statements are equivalent:
(a) f is continuous on RP.
(b) If G is open in Rº, then f-G) is open in Rº.
(c) If H is closed in Rº, then (H) is closed in RP.
![16.D. Let f and g be continuous functions on Rº to R and let h, k be defined
for x in Rº by
h(x)
= sup {f(x), g(x)},
k(x)
inf {f(x), g(x)}.
Show that h and k are continuous on R. [Hint: use the relations sup {a, b}
} {a + 6+ la – b|}, inf {a, b} = } {a + b – la - bl}.]
16.E. Let f be continuous on R? to Rº. Define the functions g., g2 on R to R by
gi (t) = f(t, 0),
92(t) = f(0, t).
Show that
91
and
92 are continuous.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6de8529-bcc7-4d6a-82fd-d13957834b62%2F34cfdc06-fc00-4702-8c25-6e307991e1ff%2Fjrrd38o_processed.png&w=3840&q=75)
Transcribed Image Text:16.D. Let f and g be continuous functions on Rº to R and let h, k be defined
for x in Rº by
h(x)
= sup {f(x), g(x)},
k(x)
inf {f(x), g(x)}.
Show that h and k are continuous on R. [Hint: use the relations sup {a, b}
} {a + 6+ la – b|}, inf {a, b} = } {a + b – la - bl}.]
16.E. Let f be continuous on R? to Rº. Define the functions g., g2 on R to R by
gi (t) = f(t, 0),
92(t) = f(0, t).
Show that
91
and
92 are continuous.
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