Here is a strange continuous function on the unit interval,
whose derivative is 0 almost everywhere, but it somehow
magically rises from 0 to 1!
Take any number X in the unit interval, and
express it in base 3. Chop off the base 3 expansion
right after the first 1. Next change all 2's in the
expansion to 1's. This number now has only 0's or 1's
in its expansion, so we can interpret it as a base 2 number!
Call this new number f(x).
If you plot this function, you get something called
the Devil's Staircase. It is related to the
standard Cantor set in the following way. This
function is constant at all the removed intervals from
the standard Cantor set. For instance if x is in
[1/3,2/3], then f(x)=1/2. If x is in [1/9,2/9], then
f(x)=1/4; if x is in [7/9,8/9], then f(x)=3/4.
If you plot this you will see that this function is
not differentiable at the Cantor set points, but has
zero derivative everywhere else! But since a
Cantor set has measure zero, this function is
has zero derivative practically everywhere, and only
"rises" on Cantor set points!
Draw a good approximation to the devil's staircase.
The Math Behind the Fact:
Pathological functions such as this one are important
and fun examples that are studied in a real analysis course.
Another fun pathology is a
Continuous but Nowhere Differentiable function.
How to Cite this Page:
Su, Francis E., et al. "Devil's Staircase."
Math Fun Facts.