applet is designed to help students
develop a visual intuition about the meaning of continuity for
functions of one variable.
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.
proving continuity requires us
to produce a rule for all positive ε, we often will be
if we can find an ε when δ has values .1, .01,
on the button below to launch the
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
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
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.
Things to notice:
Sometimes it is hard to
understand continuity since almost all
functions we can think of are continuous almost everywhere.
Consider the function f(x,y) = sign(x). It is
continuous everywhere except at the origin. Compare what
at the origin with what happens a small
distance from the origin.
When we zoom in on any
reasonable function, it eventually
approximates a line. For a line delta works as long as
epsilon/delta is greater than the slope of the line. How does
this help you find a general rule for sufficiently small epsilon with