How to Graph y = x/x in Latex

Update: For a more general solution, see this answer.

You can just overlay a small circle at the appropriate coordinate:

I plotted the axes after the circle to make the y-axis visible inside the hole.

\documentclass{article}

\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
  \draw[line width=1] (-3,1) -- (3,1);
  \draw[line width=1,fill=white] (0,1) circle (.8mm);
  \draw[->] (-3, 0) -- (3, 0) node[right] {$x$};
  \draw[->] (0, -3) -- (0, 3) node[above] {$y$};
  \node at (2,1.4) {$f(x)=\frac{x}{x}$};
  \node[below left] at (0,0) {0};
  \node[below left] at (0,1) {1};
\end{tikzpicture}

\end{document}

\documentclass[border=1 cm,tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
\begin{tikzpicture}
\begin{axis}
  \addplot[blue, thick,  mark=none] {x/x};
  \draw[blue, thick, fill=white] (0,1) circle[radius=2pt];
\end{axis}
\end{tikzpicture}
\end{document}