You've seen all sorts of functions in calculus. Most of
them are very nice and smooth they're "differentiable",
i.e., have derivatives
defined everywhere. Some, like the absolute value function,
have "problem points" where the derivative is not defined.
But is it possible to construct a
continuous function that has "problem points" everywhere?
Surprisingly, the answer is yes! Weierstrass
constructed the following example in 1872, which came as
a total surprise. It is a continuous, but nowhere
differentiable function, defined as an infinite series:
f(x) = SUM_{n=0 to infinity}
B^{n} cos(A^{n}*Pi*x)
where A and B can be any numbers such that B is between
0 and 1, and A*B is bigger than 1+(3*Pi/2). For instance,
A=12, B=1/2 will work.
Presentation Suggestions:
Draw graphs of the first few terms in the series.
The discontinuities come from the fact that the terms
wiggle faster and faster as n gets larger. But the
diminishing amplitude of the terms makes the
series converge everywhere.
The Math Behind the Fact:
Showing this infinite sum of functions (i) converges,
(ii) is continuous, but (iii) is not differentiable
is usually done in an interesting course called
real analysis
(the study of properties of real numbers and functions).
Property (ii) follows from the fact that this
series exhibits uniform convergence,
and in real analysis it is shown that
a sequence of continuous functions that converges uniformly
must converge to a continuous function.
How to Cite this Page:
Su, Francis E., et al. "Continuous but Nowhere Differentiable."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
