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.
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
f(x)=abs(x) or f(x)=x*sin(1/x), function that are continuous but
GeoGebra is a GNUed software package for mathematics visualization.
The home for the applications is http://www.geogebra.at.