# Convergence and continuity of a function

- Topic:
- Continuity

This applet illustrates the ε-δ definitions of the limit and continuity of a function.

Enter a rule for the function in the box provided.
Drag the purple point on the x-axis to adjust , and the brown point on the y-axis to adjust and continuity .
Some interesting functions to try:

**L**. Click '**Show ε**' or '**Show δ**' to display regions for**ε**and**δ**. Drag the edges of the orange or blue regions to adjust ε or δ. For each**ε**, can you find a**δ**so that all of the curve in the blue region is also in the orange region? Zoom in or out using the buttons, if needed. Use the checkboxes at bottom-left to switch between convergence- A function with a single point discontinuity: f(x) = If[ 0.99 < x < 1.01, 2, x ] Does the limit exist at x = 1? Is it continuous at x = 1?
- The Dirichlet function
. This function is built in as d(x). Where, if anywhere, does the limit exist? - A variation on the Dirichlet function:
. Enter this as f(x) = d(x)*x^2 + 1 Where does the limit of this function exist? Where is it continuous?