## Limits, Continuity, and Differentiability

This applet is designed to allow a visual exploration of the relationship between differentiability and continuity

When we say a function is continuous at x=a, we are claiming that for any height  ε > 0, we can find a width δ so that a box centered at (a,f(a)) traps the function.  We then try to find values of δ for various values of ε.

When we say a function is differentiable at x=a, we are claiming that there is a slope d, and for any ε > 0, we can find a δ  0 so that a cone (bow tie) centered at the point (a,f(a)) with slope d, where the function is trapped in a cone.  We can similarly try to find δ for various values of ε.

 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now) Min X = Max X = Min Y = Max Y = Center X = Center Y = width = height = Mike May, S.J., 2/18/2006, Created with GeoGebra

The applet lets you drag the window and zoom in or out.
You can cahnge the exampel with the command line, using commands like "f(x) = sin(x)", "P=(2,3)", and P1=(4,3)".
Some interesting curves to examine:
• f(x)= x^2-2x-1, or f(x) = sin(x), nice examples that are differentiable and continuous everywhere..
• f(x)= x/abs(x) and f(x)= sin(1/x), functions that are not continuous.
• f(x)=abs(x) or f(x)=x*sin(1/x), function that are continuous but not differentiable.

GeoGebra is a GNUed software package for mathematics visualization.  The home for the applications is  http://www.geogebra.at.