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.

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.

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Last updated By Mike May, S.J., February 29, 2008.