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In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion.

Contents 1 Examples and workarounds 1.1 Zero to the zero power 1.2 Analysis 1.3 Measure theory 2 Notation using ↓ and ↑ 3 See also

The following expressions are undefined in all contexts, but remarks in the analysis section may apply.

The following are defined in some, but not all contexts, as described in sections of this article.

	See division by zero.
00	zero to the zero power, analysis, and set theory
	analysis and set theory
	analysis and set theory
	analysis, set theory, and measure theory

Further information: Exponentiation: Zero to the zero power

The question of 0⁰ may be the most common point on which branches of mathematics disagree. Here we note only two considerations, one from analysis and one from combinatorics, as an example of the way different approaches may yield different answers.

In 1821, Cauchy also listed 0⁰ as undefined. The function 0^x (for x>0) is constantly 0, and the function x⁰ (for x>0) is constantly 1, so there seems to be no natural value for 0⁰. Indeed, for suitably chosen continuous functions f and g with whose limit as is 0 (with f taking positive values), the limit

can be any nonnegative number, or infinity, or fail to exist.

Modern textbooks often define 0⁰ = 1. For example, Ronald Graham, Donald Knuth and Oren Patashnik argue in their book Concrete mathematics:

“

Some textbooks leave the quantity 00 undefined, because the functions 0x and x0 have different limiting values when x decreases to 0. But this is a mistake. We must define x0 = 1 for all x , if the binomial theorem is to be valid when x = 0 , y = 0, and/or x = −y . The theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant.

”

In mathematical analysis the domain of a function is usually determined by the limit of the function, so as to make the function continuous. This definition makes all of the expressions undefined. In calculus, some of the expressions arise in intermediate calculations, where they are called indeterminate forms and dealt with using techniques such as L'Hôpital's rule.

In measure theory (which the common way of treating probability theory in mathematics), measures are preserved under countable addition. Taking as countable, .

In computability theory, if f is a partial function on S and a is an element of S, then this is written as f(a)↓ and is read "f(a) is defined."

If a is not in the domain of f, then f(a)↑ is written and is read as "f(a) is undefined" .

• Well-defined
• Indeterminate
• Indefinite
• Mathematical singularity
• L'Hôpital's rule
• Bottom type