Extended Real Number Line - Miscellaneous

Miscellaneous

Several functions can be continuously extended to R by taking limits. For instance, one defines exp(−∞) = 0, exp(+∞) = +∞, ln(0) = −∞, ln(+∞) = +∞ etc.

Some discontinuities may additionally be removed. For example, the function 1/x2 can be made continuous (under some definitions of continuity) by setting the value to +∞ for x = 0, and 0 for x = +∞ and x = −∞. The function 1/x can not be made continuous because the function approaches −∞ as x approaches 0 from below, and +∞ as x approaches 0 from above.

Compare the real projective line, which does not distinguish between +∞ and −∞. As a result, on one hand a function may have limit ∞ on the real projective line, while in the affinely extended real number system only the absolute value of the function has a limit, e.g. in the case of the function 1/x at x = 0. On the other hand

and

correspond on the real projective line to only a limit from the right and one from the left, respectively, with the full limit only existing when the two are equal. Thus ex and arctan(x) cannot be made continuous at x = ∞ on the real projective line.