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Exercise 5.2 Let f: (0,1) → R be such that for every x E (0,1) there exist r > 0 and a Borel measurable function g, both depending on x, such that f and g agree on (x – r,x +r) n (0,1). Prove that f is Borel measurable.

Exercise 5.2 Let f: (0,1) → R be such that for every x E (0,1) there exist r > 0 and a Borel measurable function g, both depending on x, such that f and g agree on (x – r,x +r) n (0,1). Prove that f is Borel measurable.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Need assistance with proving  that  a function f is Borel measurable for Real Analysis, Thankyou

**Exercise 5.2** Let \( f : (0,1) \to \mathbb{R} \) be such that for every \( x \in (0,1) \) there exist \( r > 0 \) and a Borel measurable function \( g \), both depending on \( x \), such that \( f \) and \( g \) agree on \( (x-r, x+r) \cap (0,1) \). Prove that \( f \) is Borel measurable.
Transcribed Image Text:**Exercise 5.2** Let \( f : (0,1) \to \mathbb{R} \) be such that for every \( x \in (0,1) \) there exist \( r > 0 \) and a Borel measurable function \( g \), both depending on \( x \), such that \( f \) and \( g \) agree on \( (x-r, x+r) \cap (0,1) \). Prove that \( f \) is Borel measurable.
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