# Is a total function also a partial function?

Is there a consensus on whether a total function, i.e., a function defined for each element of the domain, is also a partial function?

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I would guess so. In programming language semantics, a partial function $A\to B$ is understood exactly as a function $A\to B\cup \{\bot\}$, where $\bot$ is a special value that indicates failure or nontermination. So programming language researchers all think of total functions as a special case of partial functions; they are functions for which $\forall x.f(x)\ne\bot$. – MJD Aug 1 '12 at 15:59

When someone says "partial function", the usual interpretation is that the function may or may not be defined on the entire domain. The word is also sometimes used with the meaning "not total"; this is however a relatively rare meaning and will usually only be understood in contexts where the ordinary meaning would be clearly senseless.

The unambiguous way to say that a function is not total is "not total".

Note that in almost all mathematical subfields, the word "function" alone means "total function"; we only add the word "total" when there's a risk that the reader might otherwise think we were allowing non-total ones, too.

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Yes, I think generally "total" anything is usually regarded or defined as a "partial" things that happens to be total. Usually in practice, one defines the more general partial concept first and then defined the total concept afterward by adding the additional totalness condition to the original definition.

For example, you must may have seen the concept of a partial ordering. A linear ordering or total ordering is just a partial ordering in which everything is comparable.

In terms of functions, in computability theory this convention is actually used, the partial computable functions are the function given by Turing Machines but may not be defined on all $n \in \omega$. (Intuitively, algorithms or computer programs do not necessarily halt on all inputs.) Afterward, the total computable functions (or computable functions for short) are the total partial computable functions, or less akward sounding the partial computable functions that are defined for all $n \in \omega$.

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Yes. A partial function from $A$ to $B$ is a total function $U \to B$ for some $U \subseteq A$; under this definition a total function is just a partial function with $U=A$. This is valid since $A \subseteq A$.

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