# Semantics Question (function is constant v. function is a constant function v. function is a constant)

I'm definitely overthinking here and feel totally out of touch with my math skills at the moment. I'm practicing simple math problems to refresh my skills and solved a problem with ease earlier but when I tried to review it just now, I got stuck on a semantics issue (which I'll attribute to my how seriously rusty I am).

My problem says "$$X$$ is the set of all real-valued functions defined on the interval $$[a,b], a. The function $$f \in X$$ is constant on $$[a,b]$$."

Does it mean that $$f$$ is the same constant function $$\in X$$, that $$f$$ is some constant function $$\in X$$, or that practically $$f$$ can represent the set of all constant functions in $$[a,b]$$? Say, is $$f=c$$ where $$c$$ is a set constant in $$[a,b]$$, is it $$f=c$$ where $$c$$ is a some constant in $$[a,b]$$, or is it $$f=c$$ where $$c$$ is any constant in $$[a,b]$$? Does it even matter 😫? I feel stupid asking this question, bc I think the answer should be obvious, but I'm currently obsessing over it 😩🙈.

I know that regardless, the composite function $$f\circg$$ is constant for every function $$g$$ in $$X$$.

$$g$$ $$\circf$$ would also be a constant function, correct?

Would $$\lvert f(x)-f(y)\rvert\le(x-y)^2$$ for every $$x$$ and $$y$$ in $$[a,b]$$ in all cases as well?

• A constant function Is a function such that exists a Number $c$ such that $f(x) =c$ for Every $x \in [a,b]$ – astrobarrel Aug 31 at 6:53

Let $$n\geq 1$$. A function $$f:A^n\rightarrow B$$ is constant (or a constant function) means that for all $$a\in A$$, $$f(a)=b$$ for some $$b$$. I.e. the image of $$f$$ is $$\{b\}$$.
A function is a constant means that $$f: A^0\rightarrow B$$ is a $$0$$-ary function, i.e., $$f=b$$ for some $$b\in B$$. This concept is e.g. used in computability theory which is fully based on functions $$f:{\Bbb N}^n\rightarrow{\Bbb N}$$, $$n\geq 0$$. The case $$n=0$$ allows to get constants into the theory.
• The domain $A$ can be any set. But all elements of $A$ map to the same element in the codomain. – Wuestenfux Aug 31 at 10:38
The statement means $$f$$ may be any particular constant function in $$X.$$