(iii) (f/g)(x) = g(x) if g(x) #0 for all z € M. Prove that cf. fg and (f/g) are all continuous. 8. Let f and g be continuous real valued functions defined on a metric space M. Let A = {re M: f(x) < g(x)}. Prove that A is open. [Hint. A = (f-g)-¹((-∞,0))]. 9. Let M₁ and M₂ be two metric spaces and let AC M₁. If f: My → My is continuous, show that flA: A→ M₂ is also continuous.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Let M1 and M2 be two metric spaces and let A is a subset of M1. If f: M1→M2 is continuous, show that f|A: A→M2 is also continuous
Definition: Let (M,d1) and (M2, da) be metric spaces. A function
f(x)
if g(x) # 0 for all z E M.
9(x)
(iii) (F/9)(x)=
%3D
Prove that cf, fg and (f/g) are all continuous.
8. Let f and g be continuous real valued functions defined on a metrie
space M. Let A = {x € M : f(x) < g(x)}. Prove that A is open.
%3D
Hint. A = (f – g)¬'((-∞,0))].
9. Let M1 and M2 be two metric spaces and let A C M. If f: M My
is continuous, show that fla : A → M2 is also continuous.
4.3/ Homeomorphism
Definition: Ler (Mud) and (M2, dz) be metric spaces
omeomorphism if
Transcribed Image Text:Definition: Let (M,d1) and (M2, da) be metric spaces. A function f(x) if g(x) # 0 for all z E M. 9(x) (iii) (F/9)(x)= %3D Prove that cf, fg and (f/g) are all continuous. 8. Let f and g be continuous real valued functions defined on a metrie space M. Let A = {x € M : f(x) < g(x)}. Prove that A is open. %3D Hint. A = (f – g)¬'((-∞,0))]. 9. Let M1 and M2 be two metric spaces and let A C M. If f: M My is continuous, show that fla : A → M2 is also continuous. 4.3/ Homeomorphism Definition: Ler (Mud) and (M2, dz) be metric spaces omeomorphism if
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