# Convergence and continuity of a function

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 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 and continuity . Some interesting functions to try:
• 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?