continuous functions on the extended real numbers

Within this entry, ¯ will be used to refer to the extended real numbers.

Theorem 1.

Let f:RR be a function. Then f¯:R¯R¯ defined by

f¯(x)={f(x) if xA if x=B if x=-

is continuousMathworldPlanetmathPlanetmath if and only if f is continuous such that limxf(x)=A and limx-f(x)=B for some A,BR¯.


Note that f¯ is continuous if and only if limxcf¯(x)=f¯(c) for all c¯. By defintion of f¯ and the topologyMathworldPlanetmath of ¯, limxcf¯(x)=limxcf(x) for all c¯. Thus, f¯ is continuous if and only if limxcf(x)=f¯(c) for all c¯. The latter condition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( to the hypotheses that f is continuous on , limxf(x)=A, and limx-f(x)=B. ∎

Note that, without the universal assumption that f is a function from to , necessity holds, but sufficiency does not. As a counterexample to sufficiency, consider the function f¯: defined by

f¯(x)={1x2 if x{0} if x=00 if x=±.

Title continuous functions on the extended real numbers
Canonical name ContinuousFunctionsOnTheExtendedRealNumbers
Date of creation 2013-03-22 16:59:31
Last modified on 2013-03-22 16:59:31
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 12D99
Classification msc 28-00