Extended real number line

The real numbers together with positive and negative infinity, ordered so that negative infinity is below every real number and positive infinity is above every real number.

Real number: A number representing a point on the continuous number line, including rational and irrational numbers. Every ordinary member of the extended real number line is a real number.
Extended real arithmetic: Arithmetic conventions for real numbers together with positive and negative infinity. These conventions define sums, products, and order involving the endpoints.
Essential supremum: The least almost-everywhere upper bound of a measurable function, possibly positive infinity. It is a standard bound for measurable functions valued in the extended line.
Projectively extended real line: The real line with a single point at infinity, approached from both directions. Unlike this ordered line, it identifies the two directions of unboundedness.
Order topology: A topology generated by open intervals in a linearly ordered set. It gives the extended line a topology whose neighborhoods reflect its added endpoints.
Indeterminate form: A limit expression whose displayed form alone does not determine the limit’s value. Expressions such as infinity minus infinity cannot be assigned a value by endpoint arithmetic alone.
Lebesgue integral: An integral defined by measuring the sizes of a function’s level sets, allowing infinite values. Nonnegative Lebesgue integrals naturally take values in the extended real numbers.
Riemann sphere: The complex plane together with one point at infinity, forming a sphere. Its single complex infinity is not two ordered endpoints.
Limit of a function: A value that a function’s outputs approach as its input approaches a specified point or infinity. Limits on this line can take either infinite endpoint as their value.
Extended-real-valued function: A function whose values lie in the real numbers together with positive and negative infinity. Such functions use the line as their codomain.

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