
“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 <δ
0
implies |φ(t) −φ(c)| <. This means that |t −c| <δ= 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 ...