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finite limit implying uniform continuity∗
pahio†
2013-03-22 3:01:13
Theorem. If the real function f is continuous on the interval [0, ∞) and
the limit lim f (x) exists as a finite number a, then f is uniformly continuous
x→∞
on that interval.
Proof. Let ε > 0. According to the limit condition, there is a positive
number M such that
ε
2
|f (x)−a| <
∀x > M.
(1)
The function is continuous on the finite interval [0, M + 1]; hence f is also
uniformly continuous on this compact interval. Consequently, there is a positive
number δ < 1 such that
|f (x1 )−f (x2 )| < ε
∀ x1 , x2 ∈ [0, M +1] with |x1 −x2 | < δ.
(2)
Let x1 , x2 be nonnegative numbers and |x1 −x2 | < δ. Then |x1 −x2 | < 1 and
thus both numbers either belong to [0, M +1] or are greater than M . In the
latter case, by (1) we have
|f (x1 )−f (x2 )| = |f (x1 )−a+a−f (x2 )| 5 |f (x1 )−a| + |f (x2 )−a| <
ε ε
+
= ε.
2 2
(3)
So, one of the conditions (2) and (3) is always in force, whence the assertion is
true.
∗ hFiniteLimitImplyingUniformContinuityi
created: h2013-03-2i by: hpahioi version:
h41874i Privacy setting: h1i hTheoremi h26A15i
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