For questions related to contraction mapping. On a metric space $(M, d)$ it is a function $f$ from $M$ to itself, with the property that there is some real number $0\leq k<1$ such that for all $x$ and $y$ in $M$, $d(f(x),f(y))\leq k,d(x,y)$.
A contraction mapping, or contraction or contractor, on a metric space $(M, d)$ is a function $f$ from $M$ to itself, with the property that there is some real number $0\leq k<1$ such that for all $x$ and $y$ in $M$, $d(f(x),f(y))\leq k\,d(x,y)$.
More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if $(M, d)$ and $(N, d')$ are two metric spaces, then $f:M\rightarrow N$ is a contractive mapping if there is a constant $0\leq k<1$ such that $\displaystyle d'(f(x),f(y))\leq k\,d(x,y)$ for all $x$ and $y$ in $M$.
Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant $k$ is no longer necessarily less than $1$).
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