Editing Projectively extended real line (section)

== Calculus ==
The tools of [[calculus]] can be used to analyze functions of <math>\widehat{\mathbb{R}}</math>. The definitions are motivated by the topology of this space.

=== Neighbourhoods ===
Let <math>x \in \widehat{\mathbb{R}}</math> and <math>A \subseteq \widehat{\mathbb{R}}</math>.
* {{mvar|A}} is a [[Neighbourhood (mathematics)|neighbourhood]] of {{math|''x''}}, if {{math|''A''}} contains an open interval {{math|''B''}} that contains {{mvar|x}}.
* {{mvar|A}} is a right-sided neighbourhood of {{mvar|x}}, if there is a real number {{mvar|y}} such that <math>y \neq x </math> and {{mvar|A}} contains the semi-open interval <math>[x, y)</math>.
* {{mvar|A}} is a left-sided neighbourhood of {{mvar|x}}, if there is a real number {{mvar|y}} such that <math>y \neq x </math> and {{mvar|A}} contains the semi-open interval <math>(y, x]</math>.
* {{mvar|A}} is a [[punctured neighbourhood]] (resp. a right-sided or a left-sided punctured neighbourhood) of {{mvar|x}}, if <math>x\not\in A,</math> and <math>A\cup\{x\}</math> is a neighbourhood (resp. a right-sided or a left-sided neighbourhood) of {{mvar|x}}.

=== Limits ===

==== Basic definitions of limits ====
Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},</math> <math>p \in \widehat{\mathbb{R}},</math> and <math>L \in \widehat{\mathbb{R}}</math>.

The [[limit of a function|limit]] of ''f''{{hairsp}}(''x'') as {{math|''x''}} approaches ''p'' is ''L'', denoted
: <math>\lim_{x \to p}{f(x)} = L</math>
if and only if for every neighbourhood ''A'' of ''L'', there is a punctured neighbourhood ''B'' of ''p'', such that <math>x \in B</math> implies <math>f(x) \in A</math>.

The [[one-sided limit]] of ''f''{{hairsp}}(''x'') as ''x'' approaches ''p'' from the right (left) is ''L'', denoted
: <math>\lim_{x \to p^{+}}{f(x)} = L \qquad \left( \lim_{x \to p^{-}}{f(x)} = L \right),</math>
if and only if for every neighbourhood ''A'' of ''L'', there is a right-sided (left-sided) punctured neighbourhood ''B'' of ''p'', such that <math>x \in B</math> implies <math>f(x) \in A.</math>

It can be shown that <math>\lim_{x \to p}{f(x)} = L</math> if and only if both <math>\lim_{x \to p^+}{f(x)} = L</math> and <math>\lim_{x \to p^-}{f(x)} = L</math>.

==== Comparison with limits in <math>\mathbb{R}</math> ====
The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements, <math>p, L \in \mathbb{R},</math> the first limit is as defined above, and the second limit is in the usual sense:
* <math>\lim_{x \to p}{f(x)} = L</math> is equivalent to <math>\lim_{x \to p}{f(x)} = L</math>
* <math>\lim_{x \to \infty^{+}}{f(x)} = L</math> is equivalent to <math>\lim_{x \to -\infty}{f(x)} = L</math>
* <math>\lim_{x \to \infty^{-}}{f(x)} = L</math> is equivalent to <math>\lim_{x \to +\infty}{f(x)} = L</math>
* <math>\lim_{x \to p}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to p}{|f(x)|} = +\infty</math>
* <math>\lim_{x \to \infty^{+}}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to -\infty}{|f(x)|} = +\infty</math>
* <math>\lim_{x \to \infty^{-}}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to +\infty}{|f(x)|} = +\infty</math>

==== Extended definition of limits ====
Let <math>A \subseteq \widehat{\mathbb{R}}</math>. Then ''p'' is a [[limit point]] of ''A'' if and only if every neighbourhood of ''p'' includes a point <math>y \in A</math> such that <math>y \neq p.</math>

Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}}, A \subseteq \widehat{\mathbb{R}}, L \in \widehat{\mathbb{R}}, p \in \widehat{\mathbb{R}}</math>, ''p'' a limit point of ''A''. The limit of ''f''{{hairsp}}(''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 <math>x \in A \cap C</math> implies <math>f(x) \in B.</math>

This corresponds to the regular [[continuity (topology)|topological definition of continuity]], applied to the [[subspace topology]] on <math>A\cup \lbrace p \rbrace,</math> and the [[restriction (mathematics)|restriction]] of ''f'' to <math>A \cup \lbrace p \rbrace.</math>

=== Continuity ===

The function
: <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},\quad p \in \widehat{\mathbb{R}}.</math>
is [[Continuous function|continuous]] at {{math|''p''}} if and only if {{math|''f''}} is defined at {{math|''p''}} and
: <math>\lim_{x \to p}{f(x)} = f(p).</math>

If <math>A \subseteq \widehat\mathbb R,</math> the function
: <math>f : A \to \widehat{\mathbb{R}}</math>
is continuous in {{math|''A''}} if and only if, for every <math>p \in A</math>, {{math|''f''}} is defined at {{math|''p''}} and the limit of <math>f(x)</math> as {{math|''x''}} tends to {{math|''p''}} through {{math|''A''}} is <math>f(p).</math>

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

Also, if the [[tangent function|tangent]] function <math>\tan</math> is extended so that
: <math>\tan\left(\frac{\pi}{2} + n\pi\right) = \infty\text{ for }n \in \mathbb{Z},</math>
then <math>\tan</math> is continuous in <math>\mathbb{R},</math> but cannot be prolongated further to a function that is continuous in <math>\widehat{\mathbb{R}}.</math>

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

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