Is it correct to state, that bounded function, bounded in domain?

Question

I started read some further mathematics book, and continuously suspect that it is pretty low-level and contains mistakes.

In the current section I read, as the book states, that: "function $y = f(x)$ is called bounded above in the function's domain $X$, if there exists positive number $m$, so in inequality $f(x) \le m,\space ∀x ∈ X$ is true".

Is it correct to state, that it bounded (limited) in the domain (i.e. on X), but not on the range (i.e. on Y)?

Why is domain (i.e. X) related at all, if the only thing that is "restricted" is function value, i.e. range (i.e. Y)?

Every other source I found, formulates, that such function is bounded in its range, not domain.

Or is it not important, and I'm being picky?

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According to Wikipedia, (originally Jeffrey, Alan. Mathematics for Engineers and Scientists, 5th Edition):

A function $f$ defined on some set $X$ with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number $M$ such that $|f(x)|\le M$ for all x in X

So, function bound is not related, does not depend anyhow on function's domain ($X$). I think it could been confused with local maximum and minimum.

Answer 1

Short answer

No, it is not correct to state, that bounded function, bounded on domain.

Prove

Function, or generally a map, can be defined in few ways:

• Analytically: using algebraical expression and specifying domain (x) and range (y) set, for example: $Y:y=f(x),$ for $x∈X⊆\mathbb{R}, y∈Y⊆X$;
• By explicitly determining domain and range, for example: $f: X\rightarrow Y$, $X = \{1,2,3\}, Y=\{3,2,1\}$
• By plot

One common aspect of all ways is that domain and range sets define the function.

By changing domain set, we are redefying a function, we considering another function, either, if we considering a part of domain, that is subdomain, then we can't use bounding term, because by definition, function must be considered for all x [1].

If to consider only some x's, i.e. only part of domain, i.e. some subdomain, then we are dealing with local maximum and minimum [2].

Summary

Function is not bounding on domain, because the value that is restricted belongs to range and not to domain; by changing domain of the function, same function bounding is not changed, because function with another domain is another function, since function is defined by its domain.

Bounding function is bounding on its range (Y), and not on the domain (X).

Links

[1] - https://en.wikipedia.org/wiki/Bounded_function

[2] - https://math.dartmouth.edu/opencalc2/cole/lecture10.pdf (page 2)

Answer 2

It is very important which domain you consider. Take for example the function $f(x) = x$ for $x\in\mathbb{R}$. If we restrict $f$ to, let's say, $[-1,1]$, then it's bounded from above by $1$, since $f(x) = x \leq 1$ for all $x\in[-1,1]$. If we consider the domain to be $\mathbb{R}$ instead, then $f$ is clearly not bounded as $f(x) = x \rightarrow\infty$ as $x\rightarrow\infty$. Hope this clarifies your problem.