Exercise 5. Consider a subset of A of the topological space X. The characteristic function XA : X → R is, as usual, defined by XA(x) = [1, if x € A; 10, if x & A. 1 The function XA is continuous at each point in Int(A) U Ext(A)=X\A and is not continuous at any point in the boundary 8A of A. In particular, if A is dense in X and has an empty interior then XA is nowhere continuous.
Exercise 5. Consider a subset of A of the topological space X. The characteristic function XA : X → R is, as usual, defined by XA(x) = [1, if x € A; 10, if x & A. 1 The function XA is continuous at each point in Int(A) U Ext(A)=X\A and is not continuous at any point in the boundary 8A of A. In particular, if A is dense in X and has an empty interior then XA is nowhere continuous.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Exercise 5
Transcribed Image Text:Exercise 5. Consider a subset of A of the topological space X. The characteristic function XA :
X → R is, as usual, defined by
XA(x) =
[1, if x € A;
0, if x # A.
1
XOA and is not continuous at
The function XA is continuous at each point in Int(A) U Ext(A)
any point in the boundary OA of A.
In particular, if A is dense in X and has an empty interior then XA is nowhere continuous.
=
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