In this wiki a monotone map is defined, but in this paper in theorem 1.1 the definition of a monotone function is recalled. The first is concerned with points of the image, but the second is about points of the whole target space. What is the more standard term for the definition in the wiki and in the paper? Why is the term continuous monotone map used in the paper when the wiki says that monotone maps are continuous? Is this term redundant or are monotone maps not necessarily continuous?
The definition of “monotone” in the wiki article is for the very general case of continuous maps between arbitrary topological spaces. For ordinary mortals like you and me, monotonicity is defined mostly for real-valued functions of a real variable. In this world, a monotone function is either order-preserving or order-reversing. In symbols, if $X$ and $Y$ are subsets of $\mathbb R$ and $f\colon X\to Y$ is a function (not assumed to be continuous!), one says that $f$ is monotone increasing if, for $x,x'\in X$, $x<x'$ implies $f(x)<f(x')$; monotone nondecreasing if instead the conclusion is $f(x)\le f(x')$. I’ll leave it to you to guess the definitions of monotone decreasing and nonincreasing.
So, the squaring function is monotone increasing when restricted to the nonnegative numbers, $[0,\infty\rangle$, and the floor function $x\mapsto\lfloor x\rfloor$, is monotone nondecreasing, as @Tunococ has observed. Remember that $\lfloor x\rfloor$ is defined to be the largest integer less than or equal to $x$. It’s a discontinuous monotone function.