\documentclass{standalone}
\pagestyle{empty}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[scale=2]

\draw[thick]  (-2,0)--(2,0);

\draw[red] (-1,0)--(-1,1);
\draw[red] plot [parametric, domain=1:2] function {t-2*sqrt(t), t};

\draw[cyan] (2,2)--(0,0)--(-1,1);

\node at (-1.5,.5) {$u=1$};
\node at (1.5,.5) {$u=1$};
\node at (0,.5) {$u=\frac{x}{t}$};


\node at (-.65,.25) {$u=-1$};

\begin{scope}
\clip plot [parametric, domain=1:2] function {t-2*sqrt(t), t}--(2,2)--(2,0)--(-1,0);
\foreach \j in {2,1.5,...,-2} \draw [very thin,cyan] (0,0)--(\j, 2);
\foreach \j in {0,.5,...,2} \draw [very thin,cyan] (\j,0)--(\j+2, 2);
\foreach \j in {0,-.5,...,-1} \draw [very thin,cyan] (\j,0)--(\j-2, 2);
\end{scope}


\begin{scope}
\clip plot [parametric, domain=1:2] function {t-2*sqrt(t), t}--(-2,2)--(-2,0)--(-1,0);
\draw [very thin,cyan] (-1.5,0)--(.5, 2);
\draw [very thin,cyan] (-2,0)--(0, 2);
\draw [very thin,cyan] (-2.29,0)--(-.29, 2);
\draw [very thin,cyan] (-2.67,0)--(-.67, 2);
\draw [very thin,cyan] (-3.17,0)--(-1.17, 2);
\draw [very thin,cyan] (-3.67,0)--(-1.67, 2);


\end{scope}



\end{tikzpicture}
\end{document}

