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Let $f:X \to Y$ a function. Then, $$\forall x_1, x_2 \in X.~x_1=x_2 \implies f(x_1) = f(x_2)$$

What would this property be called?

gust
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    This just means that the function is "well defined". That is to say, if $f$ satisfies this property, then $f$ is well-defined. – Arun Bharadwaj Feb 19 '21 at 01:51
  • Here's a related post that exemplifies the answer very nicely: https://math.stackexchange.com/questions/606917/well-defined-function-what-does-it-mean – Arun Bharadwaj Feb 19 '21 at 01:56

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Informally, a function is a rule that assings a number to a specific output. Suppose that $x_1$ and $x_2$ were the same number, but produced a different output. Then, as Arun has mentioned in the comments, we wouldn't even be dealing with a function. For instance, consider the square root function $S:x\mapsto \sqrt{x}$. In order for this function to be well-defined, we have to designate $\sqrt{x}$ as the nonnegative square root of $x$. Otherwise, $S$ would output not a single number, but a set of numbers*: $$ S(x^2)=\{x,-x \} \, . $$ The formal definition of a function $f$ is that it is a binary relation between two sets $X$ and $Y$. The graph of a function $G$ refers to the following set $$ G(f)=\{(x,f(x)) \mid x \in X\} \, . $$ In order for a function to be well-defined, we require that if $(x,y)$ and $(x,z)$ are members of $G$, then $y$ must be equal to $z$. Hence, all functions must have the property that the same input produces the same output, otherwise we wouldn't be dealing with a function.

*In complex analysis, sometimes we relax the rules a little, and allow a function to have multiple outputs. For instance, we would allow $S$ to associate a number with its two square roots: $$ S(-1)=\{i,-i\} $$ This is called a 'multifunction'. It's worth noting, however, that in the strict interpretation of what a function is, a 'multifunction' is not a function at all.

Joe
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