Why, generally, theorems are defined for $y$-coords and not for $x$-coords?

Question

My question is specific to this theorem, but I have noticed that it applies to many others.

The Bolzano's theorem states:

Let $f$ a continuous function in $[a, b]$ and if $f(a) \cdot f(b) < 0$, therefore there are at least one point $x \in (a, b)$ such that $f(x) = 0$

But, why not state a theorem for $x$-coords? I mean to something like that:

Let $f$ a continuos function in $[a, b]$ and if $a \cdot b < 0$, therefore exist one point $y \in [min(f(k)), max(f(k))]$ with $k \in (a,b)$ such that $f(0) = y$

I don't know if it has to do, but another example would be that the definition of limit $\epsilon-\delta$ that is defined to guarantee an $y$-coord(again the $y$-coord).

Answer 1

Ultimately, I think the point is that for functions of a single variable, the longstanding convention is that $x$ is the independent variable and $y$ is the dependent variable. Of course, we usually substitute $y = f(x)$ to underscore this dependence. One can see this relationship in the $\varepsilon$-$\delta$ definition (of limits, continuity, etc.): the $y$-value $\varepsilon$ is fixed, and the thrust is to find a real number $\delta > 0$ that measures the closeness of $x$ to some value $a$ in order to guarantee the closeness of $f(x)$ to some value $L$ or $f(a).$ Likewise, the theorem that you state in your question can be rephrased as follows:

Given a continuous function $f(x)$ on a closed interval $[a, b],$ if we have that $f(x) \neq 0$ for any $a < x < b,$ then $f(x)$ does not change sign for any $a < x < b.$

Once again, the dependence of $f(x)$ on the variable $x$ comes into play.