Permissable to define a fork function with two x values mapping to different functions but whose y-values are identical? Say we got this simple example
$$f(x) = \begin{cases}
p_1(x),\text{ for }5 \leq x \leq 8,\\
p_2(x),\text{ for }8 \leq x \leq 10
\end{cases}$$
Assuming $p_1(8)=p_2(8)$, is this permissable?
Arguments for and against:

*

*Yes, because $f(x)$ maps each x-value to exactly one y-value.

*No, because it becomes unclear whether to choose $p_1$ or $p_2$ when $x=8$.

*

*Counter: well, it doesn't matter what function you choose, they yield identical y-values.



 A: I would say it is an acceptable definition, in the sense its formalization describes a function. However, some people may be displeased, but again some people will always be displeased...
A function $f:X\rightarrow Y$ is a subset $F$ of $X\times Y$ such that:

*

*for every $x\in X$, there exists a $y\in Y$ for which $(x,y)\in F$;

*and, if $(x, y_{1}), (x, y_{2})\in F$, then $y_{1}=y_{2}$.

When we define a function by cases, that is not absolutely precise in the sense of this definition, but the correspondent idea can always be formalized: in your case, the set $F$ corresponding to the function may be defined as the subset of $[5, 10]\times\mathbb{R}$ (I am assuming here that your function is real-valued and not defined outside of $[5,10]$) such that
$$F=\{(x,y)\in[5,10]\times\mathbb{R} : \text{if $x\in[5,8]$, $y=p_{1}(x)$, and if $x\in [8,10]$, $y=p_{2}(x)$}\}.$$
Does that define a function?

*

*For every $x\in [5,10]$, there indeed exists an $y\in\mathbb{R}$ such that $(x,y)\in F$, equal to $p_{1}(x)$ if $x\in[5,8)$, equal to $p_{2}(x)$ if $x\in(8,10]$, and equal to $p_{1}(x)=p_{2}(x)$ if $x=8$.

*If $(x, y_{1}), (x, y_{2})\in F$, there are three cases: if $x\in[5,8)$, since $p_{1}$ is a function $y_{1}=y_{2}$; if $x\in (8,10]$, since $p_{2}$ is a function $y_{1}=y_{2}$; and if $x=8$, given both $p_{1}$ and $p_{2}$ are functions and $p_{1}(8)=p_{2}(8)$, $y_{1}=y_{2}$.

So $F$ indeed describes a function, and your definition is indeed acceptable.
