Functions have by definition the property that if $f(x_1)=a$ and $f(x_1)=b$, then $a=b$. What is the name of this property?

Question

Does this property have a name?

Answer 1

(The relation is) "uniquely defined" or "tubular at $X$" where $X$ is the left side of the relation. See this wiki page. The former seems preferred by many, including me.

As @RobArthan indicates, there's yet another name: (The relation is) "functional". See this site. The other property for a relation to be a function (fully defined) is called "entire".

FYI:

A function is a uniquely and fully defined relation.

A relation $R$ is well defined $\Leftrightarrow$ $R$ is a function $\Leftrightarrow$ $R$ is both uniquely and fully defined.

"Uniquely defined" is called "rechtseindeutig" in German, which is literally "right unique".

"Fully defined" is called "linkstotal" in German, which is "left total".

Answer 2

This is just inherent in the definition of a function, which cannot take one input to two distinct outputs.

As for a name for it, I think it's often referred to as a function being "well-defined."

Answer 3

In the standard set-theoretic way of defining functions, a function is a binary relation (i.e., a subset of $X \times Y$ for some sets $X$ and $Y$) with the property that $(x,y),(x,y') \in R \implies y = y'$, for all $(x,y)$ and $(x,y')$. So the property you mention is just called "is a function".

Answer 4

A typical English-Maths speaking (or English-Math speaking) audience will immediately understand "R is a functional relation" or "R is a single-valued relation" or (as Peter suggests) just "R is a function". As a native English-Maths speaker, I would not understand "R is tubular", and would only understand "R is right-unique" because I know the German "richtseindeutig".