Projectively extended real line

The arithmetic operations on this space are an extension of the same operations on reals. A motivation for the new definitions is the limits of functions of real numbers.

In addition to the standard operations on the subset ${\displaystyle \mathbb {R} }$ of ${\displaystyle {\widehat {\mathbb {R} }}}$, the following operations are defined for ${\displaystyle a\in {\widehat {\mathbb {R} }}}$, with exceptions as indicated:[3][2]

${\displaystyle {\begin{aligned}a+\infty =\infty +a&=\infty ,&a\neq \infty \\a-\infty =\infty -a&=\infty ,&a\neq \infty \\a/\infty =a\cdot 0=0\cdot a&=0,&a\neq \infty \\\infty /a&=\infty ,&a\neq \infty \\a/0=a\cdot \infty =\infty \cdot a&=\infty ,&a\neq 0\\0/a&=0,&a\neq 0\end{aligned}}}$

The following expressions cannot be motivated by considering limits of real functions, and no definition of them allows the statement of the standard algebraic properties to be retained unchanged in form for all defined cases.[lower-alpha 1] Consequently, they are left undefined:

${\displaystyle {\begin{aligned}&\infty +\infty \\&\infty -\infty \\&\infty \cdot 0\\&0\cdot \infty \\&\infty /\infty \\&0/0\end{aligned}}}$

The exponential function ${\displaystyle e^{x}}$ cannot be extended to ${\displaystyle {\widehat {\mathbb {R} }}}$.[2]

Let ${\displaystyle x\in {\widehat {\mathbb {R} }}}$ and ${\displaystyle A\subseteq {\widehat {\mathbb {R} }}}$.

• A is a neighbourhood of x, if A contains an open interval B that contains x.
• A is a right-sided neighbourhood of x, if there is a real number y such that ${\displaystyle y\neq x}$ and A contains the semi-open interval ${\displaystyle [x,y)}$.
• A is a left-sided neighbourhood of x, if there is a real number y such that ${\displaystyle y\neq x}$ and A contains the semi-open interval ${\displaystyle (y,x]}$.
• A is a punctured neighbourhood (resp. a right-sided or a left-sided punctured neighbourhood) of x, if ${\displaystyle x\not \in A,}$ and ${\displaystyle A\cup \{x\}}$ is a neighbourhood (resp. a right-sided or a left-sided neighbourhood) of x.

Let ${\displaystyle f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},}$ ${\displaystyle p\in {\widehat {\mathbb {R} }},}$ and ${\displaystyle L\in {\widehat {\mathbb {R} }}}$.

The limit of f (x) as x approaches p is L, denoted

${\displaystyle \lim _{x\to p}{f(x)}=L}$

if and only if for every neighbourhood A of L, there is a punctured neighbourhood B of p, such that ${\displaystyle x\in B}$ implies ${\displaystyle f(x)\in A}$.

The one-sided limit of f (x) as x approaches p from the right (left) is L, denoted

${\displaystyle \lim _{x\to p^{+}}{f(x)}=L\qquad \left(\lim _{x\to p^{-}}{f(x)}=L\right),}$

if and only if for every neighbourhood A of L, there is a right-sided (left-sided) punctured neighbourhood B of p, such that ${\displaystyle x\in B}$ implies ${\displaystyle f(x)\in A.}$

It can be shown that ${\displaystyle \lim _{x\to p}{f(x)}=L}$ if and only if both ${\displaystyle \lim _{x\to p^{+}}{f(x)}=L}$ and ${\displaystyle \lim _{x\to p^{-}}{f(x)}=L}$.

The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements, ${\displaystyle p,L\in \mathbb {R} ,}$ the first limit is as defined above, and the second limit is in the usual sense:

• ${\displaystyle \lim _{x\to p}{f(x)}=L}$ is equivalent to ${\displaystyle \lim _{x\to p}{f(x)}=L}$
• ${\displaystyle \lim _{x\to \infty ^{+}}{f(x)}=L}$ is equivalent to ${\displaystyle \lim _{x\to -\infty }{f(x)}=L}$
• ${\displaystyle \lim _{x\to \infty ^{-}}{f(x)}=L}$ is equivalent to ${\displaystyle \lim _{x\to +\infty }{f(x)}=L}$
• ${\displaystyle \lim _{x\to p}{f(x)}=\infty }$ is equivalent to ${\displaystyle \lim _{x\to p}{|f(x)|}=+\infty }$
• ${\displaystyle \lim _{x\to \infty ^{+}}{f(x)}=\infty }$ is equivalent to ${\displaystyle \lim _{x\to -\infty }{|f(x)|}=+\infty }$
• ${\displaystyle \lim _{x\to \infty ^{-}}{f(x)}=\infty }$ is equivalent to ${\displaystyle \lim _{x\to +\infty }{|f(x)|}=+\infty }$

Let ${\displaystyle A\subseteq {\widehat {\mathbb {R} }}}$. Then p is a limit point of A if and only if every neighbourhood of p includes a point ${\displaystyle y\in A}$ such that ${\displaystyle y\neq p.}$

Let ${\displaystyle f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},A\subseteq {\widehat {\mathbb {R} }},L\in {\widehat {\mathbb {R} }},p\in {\widehat {\mathbb {R} }}}$, p a limit point of A. The limit of f (x) as x approaches p through A is L, if and only if for every neighbourhood B of L, there is a punctured neighbourhood C of p, such that ${\displaystyle x\in A\cap C}$ implies ${\displaystyle f(x)\in B.}$

This corresponds to the regular topological definition of continuity, applied to the subspace topology on ${\displaystyle A\cup \lbrace p\rbrace ,}$ and the restriction of f to ${\displaystyle A\cup \lbrace p\rbrace .}$

The function

${\displaystyle f:{\widehat {\mathbb {R} }}\to {\widehat {\mathbb {R} }},\quad p\in {\widehat {\mathbb {R} }}.}$

is continuous at p if and only if f is defined at p and

${\displaystyle \lim _{x\to p}{f(x)}=f(p).}$

If ${\displaystyle A\subseteq {\widehat {\mathbb {R} }},}$ the function

${\displaystyle f:A\to {\widehat {\mathbb {R} }}}$

is continuous in A if and only if, for every ${\displaystyle p\in A}$, f is defined at p and the limit of ${\displaystyle f(x)}$ as x tends to p through A is ${\displaystyle f(p).}$

Every rational function P(x)/Q(x), where P and Q are polynomials, can be prolongated, in a unique way, to a function from ${\displaystyle {\widehat {\mathbb {R} }}}$ to ${\displaystyle {\widehat {\mathbb {R} }}}$ that is continuous in ${\displaystyle {\widehat {\mathbb {R} }}.}$ In particular, this is the case of polynomial functions, which take the value ${\displaystyle \infty }$ at ${\displaystyle \infty ,}$ if they are not constant.

Also, if the tangent function ${\displaystyle \tan }$ is extended so that

${\displaystyle \tan \left({\frac {\pi }{2}}+n\pi \right)=\infty {\text{ for }}n\in \mathbb {Z} ,}$

then ${\displaystyle \tan }$ is continuous in ${\displaystyle \mathbb {R} ,}$ but cannot be prolongated further to a function that is continuous in ${\displaystyle {\widehat {\mathbb {R} }}.}$

Many elementary functions that are continuous in ${\displaystyle \mathbb {R} }$ cannot be prolongated to functions that are continuous in ${\displaystyle {\widehat {\mathbb {R} }}.}$ This is the case, for example, of the exponential function and all trigonometric functions. For example, the sine function is continuous in ${\displaystyle \mathbb {R} ,}$ but it cannot be made continuous at ${\displaystyle \infty .}$ As seen above, the tangent function can be prolongated to a function that is continuous in ${\displaystyle \mathbb {R} ,}$ but this function cannot be made continuous at ${\displaystyle \infty .}$

Many discontinuous functions that become continuous when the codomain is extended to ${\displaystyle {\widehat {\mathbb {R} }}}$ remain discontinuous if the codomain is extended to the affinely extended real number system ${\displaystyle {\overline {\mathbb {R} }}.}$ This is the case of the function ${\displaystyle x\mapsto {\frac {1}{x}}.}$ On the other hand, some functions that are continuous in ${\displaystyle \mathbb {R} }$ and discontinuous at ${\displaystyle \infty \in {\widehat {\mathbb {R} }}}$ become continuous if the domain is extended to ${\displaystyle {\overline {\mathbb {R} }}.}$ This is the case for the arctangent.