Scroll down to the bottom to view the interactive graph.


A function \(f(x,y)\) is continuous at a point \((x_0,y_0)\) if \[\lim\limits_{(x,y)\to(x_0,y_0)} f(x,y) = f(x_0,y_0).\] For example, the function \[f(x,y) = \dfrac{2xy}{x^2+y^2},\] is discontinuous at \((0,0)\) because \(f(0,0)\) is not defined. If we look more closely, however, we will notice that the function does not have a limit at \((0,0)\) either, but we could indeed find a limit no matter with direction we choose to approach to \((0,0)\).

Suppose we approach \((0,0)\) along the direction which makes an angle \(\theta\) with the \(x\)-axis. In this case the trajectory can be described by \((x,y) = (t\cos\theta, t\sin\theta)\) when \(t \to 0\). Plugging it in \(f\) gives \[f(t\cos\theta, t\sin\theta) = \dfrac{2t^2\cos\theta \sin\theta}{t^2\cos^2\theta + t^2\sin^2\theta} = \sin 2\theta.\] As \(t\to 0\), \(f \to \sin 2\theta\) which depends on \(\theta\). Since \(f\) approaches to different values along different trajectories, we say that the limit \(\lim\limits_{(x,y)\to(0,0)} f(x,y)\) does not exist.


To view the interactive graph:

  1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.
  2. Download this file, discontinuity.ggb.
  3. Click here to open GeoGebra.
  4. After you open GeoGebra, click "File" in the toolbar, then click "Open".
  5. Choose the .ggb file you just downloaded and click the "Open" button.
  6. Now you should be able to view the graph inside GeoGebra.

To rotate the graph, right click and drag.

Joseph Lo