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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 $[a,b], a<b$. 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$$\circ$$g$ is constant for every function $g$ in $X$.

$g$ $\circ$$f$ 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?

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

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  • $\begingroup$ So a function being a constant means that it has a trivial domain? I can somewhat understand the difference, but my brain keeps on fusing the concepts ↑ together, so I'm having a hard time separating one from the other :/ $\endgroup$ – Morg Man Aug 31 at 10:01
  • $\begingroup$ The domain $A$ can be any set. But all elements of $A$ map to the same element in the codomain. $\endgroup$ – Wuestenfux Aug 31 at 10:38
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The statement means $f$ may be any particular constant function in $X.$

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