
“real: chapter_12” — 2011/5/23 — 1:04 — page9—#9
Real-valued Functions of Two Real Variables 12-9
Proof By Theorem 12.3.10, for any two points a and b in D, we have
f (b) = f (a) or that f is a constant.
Corollary 12.3.12 Let f : D → R (where D is an open convex
subset of
R
2
) be a function such that
∂f
∂x
,
∂f
∂y
exist and are continuous
everywhere in D and
∂f
∂x
≤ M,
∂f
∂y
≤ M for all points in D. Then f
is uniformly continuous on D.
Proof By Theorem 12.3.10 and our hypothesis, for any a, b ∈ D
|f (b) − f (a)|≤M (|b
1
− a
1
|+|b
2
− a
2
|) ≤ 2M b − a .
In particular b − a <δ= /2M implies that |f (b) − f (a)| <
proving uniform continuity of f on D.
12.4 ...