This interactive page is intended to demonstrate the `\(\epsilon-\delta\)' definition of continuity.

If viewing this on a mobile device we suggest that you use 'landscape' mode.

In the right hand diagram a blue open ball \(B(w,\epsilon)\) is displayed with \(w\) shown in red.

In the left hand diagram there is a yellow disc with a red dot at the centre. The red dot represents the point \(z\) where \(f(z)=w.\) The yellow disc represents the open ball \(B(z,\delta)\). The yellow shape in the right hand diagram is the image of \(B(z,\delta)\) under the current function. For a continuous function it is possible to reduce \(\delta\) so that \(B(z,\delta)\) is completely within \(B(w,\epsilon)\) however small \(\epsilon\) is.

The fifth function is not continuous and \(z\) has been chosen to show that it is not possible to reduce \(\delta\) so that \(B(z,\delta)\) is completely within \(B(w,\epsilon)\).

• Zoom in or out, by pressing the appropriate buttons.
• Use the slider (or in older browsers) the drop down list to reduce the value of \(\delta\) by the chosen factor.
• Press 'Next epsilon' for a smaller value of \(\epsilon\).
• Press 'Reset' to return to the initial state.
• Choose the function from the list.