Defined and undefined

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In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible output. These are also known as "well-behaved" and respectively "ill-behaved".

The following algebraic expressions are undefined:

• <math>\frac{x}{0}</math> (see division by zero)
• <math>0^0</math> (however, it is often convenient to define the value of this expression to be 1, e.g. in combinatorics, in the context of power series, and in set theory)
• <math>\infty - \infty</math>
• <math>\infty^0</math>
• <math>1^\infty</math>
• <math>0 \cdot \pm\infty</math> (however, in probability and measure theory this is usually defined as having value 0)
• <math>\frac{\pm\infty}{\pm\infty}</math>

Undefined values of a function are usually determined by the limit of the function as it approaches the singularity.

See also: mathematical singularity, indeterminate, L'Hôpital's rule

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