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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_12” 2011/5/23 1:04 page3—#3
Real-valued Functions of Two Real Variables 12-3
Proof Fix x, y D, x = y and define φ : [0, 1]→R by φ(t) =
f (tx +(1 t)y). We first claim that φ is continuous at each c ∈[0, 1].
Indeed, by continuity of f, given >0 there exists a δ
0
> 0 such that
tx + (1 t)y
cx + (1 c)y
=|t c|x y
0
implies |φ(t) φ(c)| <. This means that |t c| = δ
0
/(x y)
implies that |φ(t) φ(c)| <or that φ is continuous at c. Since
φ(0) = f (y ) and φ(1) = f (x), by the intermediate value property of
continuous functions, φ assumes all the values between φ(0) and φ(1)
or that f assumes all values between ...
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Publisher Resources

ISBN: 9781299447561Publisher Website