\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}

\usepackage{amssymb,amsmath}
\usepackage{pgfplots}

\begin{document}
\begin{preview}
\begin{tikzpicture}
    \begin{axis}[
        axis lines=middle,
        width=15cm, height=15cm,     % size of the image
        grid = both,
        grid style={dashed, gray!30},
        enlargelimits=true,
        xmin=-1,     % start the diagram at this x-coordinate
        xmax= 1,     % end   the diagram at this x-coordinate
        ymin= 0,     % start the diagram at this y-coordinate
        ymax= 1,     % end   the diagram at this y-coordinate
        /pgfplots/xtick={-1,-0.8,...,1}, % make steps of length 0.2
        /pgfplots/ytick={0,0.1,...,1},   % make steps of length 0.1
        axis background/.style={fill=white},
        ylabel=y,
        xlabel=x,
        legend style={at={(0.8,0.8)}, anchor=north}]
      \addplot[domain=-1:1, ultra thick,samples=100,red]  {1}; 
      \addplot[domain=-1:1, ultra thick,samples=100,blue] {0}; 
      \legend{$1 \text{, if } x \in \mathbb{Q}$, $0 \text{, if } x \in \mathbb{R} \setminus \mathbb{Q}$}
    \end{axis} 
\end{tikzpicture}
\end{preview}
\end{document}