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?