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\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture} 
	\centering
	\begin{axis}[
		xlabel={y},
		ylabel={z},
		ymax=1,
         xticklabel={\ifdim\tick pt=0pt \else\pgfmathprintnumber{\tick}\fi},
		grid=major,
		domain=0:5,
		samples=1000,
             width=\textwidth,
            height = 8cm,
		axis lines=left,
             %heigth=5cm,
		legend pos=north west, 
		]
		
		\addplot[red,line width=1.5pt] {mod(\x,1)<1? mod(\x,1)/5:0};
		\addlegendentry{1};
		\addplot[blue,line width=1.5pt]{0.2*x};
		\addlegendentry{2};
		
\end{axis}

	\end{tikzpicture}
	\end{document}

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\usetikzlibrary{arrows, decorations.markings}
\makeatletter
\tikzset{
    nomorepostaction/.code=\makeatletter\let\tikz@postactions\pgfutil@empty,
    my axis/.style={
        postaction={
            decoration={
                markings,
                mark=at position 1 with {
                    \arrow[ultra thick]{latex}
                }
            },
            decorate,
            nomorepostaction
        },
        thin,
        -, % switch off other arrow tips
        every path/.append style=my axis % this is necessary
          %so it works both with "axis lines=left" and "axis lines=center"
    }
}
\makeatother
\begin{document}
\begin{tikzpicture} 
	\centering
	\begin{axis}[
		xlabel={y},
		ylabel={z},
		ymax=1,
         	xticklabel={\ifdim\tick pt=0pt \else\pgfmathprintnumber{\tick}\fi},
		grid=major,
		domain=0:5,
		samples=1000,
		width=\textwidth,
		height = 8cm,
		axis lines=left,
		axis line style={my axis},
             	%heigth=5cm,
		legend pos=north west, 
		]
		\addplot[red,line width=1.5pt] {mod(\x,1)<1? mod(\x,1)/5:0};
		\addlegendentry{1};
		\addplot[blue,line width=1.5pt]{0.2*x};
		\addlegendentry{2};
	\end{axis}
\end{tikzpicture}
\end{document}