When does a function NOT have a derivative?

For some functions, there are certain [math]x[/math]-values where the derivative of the function does not exist. Drag the slider at the bottom of the window to change the function. Drag point [math]P[/math] to explore the slope of the line through [math]P[/math] and the point where the derivative does not exist.
[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]

Information: When does a function NOT have a derivative?