Well-defined expression

In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous.^[1] A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if ${\displaystyle f}$ takes real numbers as input, and if ${\displaystyle f(0.5)}$ does not equal ${\displaystyle f(1/2)}$ then ${\displaystyle f}$ is not well defined (and thus not a function).^[2] The term well-defined can also be used to indicate that a logical expression is unambiguous or uncontradictory.

A function that is not well defined is not the same as a function that is undefined. For example, if ${\displaystyle f(x)={\frac {1}{x}}}$, then even though ${\displaystyle f(0)}$ is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of ${\displaystyle f}$.