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If ''L'' is a prefunction ''p'' : ''S'' <math>\rightharpoonup</math> ''T'' that happens to be total at ''S'', then ''L'' is called a ''function'' from ''S'' to ''T'', indicated by writing ''L'' = ''f'' : ''S'' &#8594; ''T''.  To say that a relation ''L'' &sube; ''S'' × ''T'' is ''totally tubular'' at ''S'' is to say that it is 1-regular at ''S''.  Thus, we may formalize the following definition:

If ''L'' is a prefunction ''p'' : ''S'' <math>\rightharpoonup</math> ''T'' that happens to be total at ''S'', then ''L'' is called a ''function'' from ''S'' to ''T'', indicated by writing ''L'' = ''f'' : ''S'' &rarr; ''T''.  To say that a relation ''L'' &sube; ''S'' × ''T'' is ''totally tubular'' at ''S'' is to say that it is 1-regular at ''S''.  Thus, we may formalize the following definition:

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{| cellpadding="4"

| ''L''

| ''L''

| =

| =

| ''f'' : ''S'' &#8594; ''T''

| ''f'' : ''S'' &rarr; ''T''

| if and only if

| if and only if

| ''L''

| ''L''

|}

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In the case of a function ''f'' : ''S'' &#8594; ''T'', one has the following additional definitions:

In the case of a function ''f'' : ''S'' &rarr; ''T'', one has the following additional definitions:

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