Continuity
Applet
This
applet is designed to help students
develop a visual intuition about the meaning of continuity for
functions of one variable.
 Recall
that for a function f of one
variable x, the function is continuous at x=x0 if for any positive
ε, we can
find a positive δ such that a box centered at (x0,f(x0)) going
up and down ε and right and left δ will trap the
function so it goes out of the sides rather than the top and bottom.

While
proving continuity requires us
to produce a rule for all positive ε, we often will be
convinced
if we can find an ε when δ has values .1, .01,
and .001.
Click
on the button below to launch the
applet.
 The applet has a control
window and 2 graphics windows.
 The graph window shows a
graph of the function, with a red dot at
the graph at x0 and a magenta patch on the axis for the
epsilon neighborhood of x0 and a corresponding magenta patch
on the graph. It also has a light blue box of radius
δ
at a height of ε above and below the graph of x0.
 The values of the
coordinates are on the control panel. The
values of ε , δ , and x0 can either be typed in or
shifted with the
arrow buttons.
 The close up window zooms in
on the disks. For a given
ε, you want to find a δ where the magenta curve
is trapped in the blue box.
 The tools menu on the
graphics windows lets you change the action
of a mouse drag between rotations and translations of the image.