problem with array "how to make it in the same line"

I would do that with aligned– simpler code and better spacing. A small improvement with colon in the place of a real colon (better spacing) and the \mfrac command, preferably to \dfrac for online maths:

\documentclass{article}

\usepackage{mathtools, amssymb, amsthm}
\usepackage{nccmath}

\begin{document}

let
$ \begin{aligned}[t]
f\colon \mathbb{R}&\longrightarrow ]-1,1[\\[-0.5ex]
x & \longmapsto \mfrac{x}{1+|x|}
\end{aligned}\; $
be a continuous function

\end{document} 

array has an optional argument to specify which line of the array should be placed on the baseline of the surrounding text, so just do \begin{array}[t]{cccc}. t for top, aligns the first row of the array with the text.

\documentclass{article}

\usepackage{amsmath,amssymb,amsthm} 


\begin{document}

let 
$
\begin{array}[t]{cccc}
f: & \mathbb{R}&\longrightarrow &]-1,1[\\
&x&\mapsto & \dfrac{x}{1+|x|}
\end{array}
$ 
be a continuous function 

\end{document}

Considering that the tag is also arrays (peraphs is off-topic with lot of sincerity) you could use also tikz-cd or a matrix. My proposal is this MWE.

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{tikz-cd}
\usepackage{amssymb}
\usepackage{lipsum}

\begin{document}
let \begin{tikzcd}[column sep=.15in,row sep=-.1in]
f\colon \mathbb{R} \arrow[r] & {]-1,1[}         \\
x \arrow[r, maps to]         & \frac{x}{1+|x|}
\end{tikzcd} be a continuous function \lipsum[1]
\end{document}

Of course, you should take into account that the two mathematical lines affect in the space colored with the red color if there is more text.