4. Let f: RR. For each subset A CR, we define its preimage by f-¹(A) = { € R" | f(x) = A}. Prove that f is continuous if and only if for every open set OCR, f-¹(0) is open.
4. Let f: RR. For each subset A CR, we define its preimage by f-¹(A) = { € R" | f(x) = A}. Prove that f is continuous if and only if for every open set OCR, f-¹(0) is open.
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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Transcribed Image Text:4.
Let f: RR. For each subset ACR, we define its preimage
by f-¹(A) = { € R" | f(x) = A}.
Prove that f is continuous if and only if for every open set OCR,
f-¹(0) is open.
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