Convergence and continuity of a function
This applet illustrates the εδ definitions of the limit and continuity of a function.
Enter a rule for the function $f(x)$ in the box provided.
Drag the purple point on the xaxis to adjust $x_0$, and the brown point on the yaxis to adjust L.
Click Show ε or Show δ to display regions for ε and δ. Drag the edges of the orange or blueregions 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 bottomleft to switch between convergence $\Bigl( \displaystyle \lim_{x \to x_0} f(x) = L \Bigr)$ and continuity $\Bigl( \displaystyle \lim_{x \to x_0} f(x) = f(x_0) \Bigr)$.
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 $\begin{cases} 1 & \text{ if } x \in \mathbb{Q} \\ 0 & \text{ if } x \notin \mathbb{Q}\end{cases}$. Enter this function as
f(x) = d(x)
Where, if anywhere, does the limit exist? 
A variation on the Dirichlet function: $\begin{cases} x^2 + 1 & \text{ if } x \in \mathbb{Q} \\ 1 & \text{ if } x \notin \mathbb{Q}\end{cases}$.
Enter this as
f(x) = d(x)*x^2 + 1
Where does the limit of this function exist? Where is it continuous?
Other resources: