This applet illustrates the concept of differentiability.

The upper section shows the graph of , and a chord from a point (, ()) to (, ()).
Drag the points and to change them. What do you get as ?
Recall that the derivative of at the point is defined as
that is, as a limit of the chord gradient function .
Click 'Show chord slope curve' to show in the lower pane a plot of the chord gradient function , as a function of .
Click 'Show derivative' to add a plot of . What happens to and at ?
Some interesting functions to try:

The absolute value function |x|: enter this as f(x) = abs(x)
Is this function differentiable at x = 0? At x ≠ 0?

A function with a single point discontinuity: f(x) = If[ 0.98 < x < 1.02, 2, x ]
Is the function differentiable at x = 1?

A function with a step discontinuity: f(x) = If[x < 1, x, x+1]
Is the function differentiable at x = 1?

The Dirichlet function . This function is built in as d(x).
Is this function differentiable?

A variation on the Dirichlet function: . Enter this as f(x) = d(x)*x^2 + 1
Where is this function differentiable?