\begin{tikzpicture}

\begin{axis}[
 x=1cm,y=1cm,
axis lines=middle,
      axis x line=middle,
        axis y line=middle,
        %enlarge x limits=0.15,
        %enlarge y limits=0.15,
        every axis x label/.style={at={(current axis.right of origin)},anchor=north east},
        every axis y label/.style={at={(current axis.above origin)},anchor=north east},
xmin=-0.5,
xmax=6.,
ymin=-0.5,
ymax=5,
ytick={1,...,2},
xtick={3},
xlabel=$x$,
ylabel=$y$,
]

\addplot[domain=0.75:3, blue!80!black, line width=1,samples=50] {3/x};
\addplot[domain=3:5, blue!80!black, line width=1,samples=50] {x-1};
\addplot[color=blue!80!black, only marks, style={mark=*, fill=white}] coordinates {(3,2)};
\addplot[color=blue!80!black, only marks, style={mark=*	}] coordinates {(3,1)};

\draw [dashed, draw=black] (axis cs: -0.05,1) -- (axis cs: 3,1);
\draw [dashed, draw=black] (axis cs: -0.05,2) -- (axis cs: 3,2);
\draw [dashed, draw=black] (axis cs: 3,-0.05) -- (axis cs: 3,2);

\node[] at (axis cs:4.5,4.5) {\footnotesize$y=\left\{\begin{array}{l}
\frac{3}{x} \text { für } 0<x \leq 3 \\
x-1 \text { für } x>3
\end{array}\right.$};

\end{axis}
\end{tikzpicture}