, nowhere differentiable function
may be constructed as follows:
define ψ(x)=|x| for x ∈ [-1,1]
extend ψ to the whole real line by requiring that ψ(x+2)=ψ(x) (i.e. ψ(x) is periodic)
This function is continuous on the whole of the real line, but nowhere differentiable (proof of this is left as an exercise for the reader). Weierstrass constructed this function in 1860. Another such function may be found in the node on differentiable functions or here.
To the uninitiated reader this may seem like merely another mathematical curiosity, to one more acquainted with mathematics it may serve as a reminder that analysis is an area where one must tread carefully, but this function played a far more important role in history.
Until then mathematicians had done analysis in a very intuitive way. They looked at pictures and drew diagrams. For example when dealing with the intermediate value theorem they might try draw some continuous functions and see that the theorem was intuitively obvious. Even today some students are tempted to draw a picture and wave their hands a bit rather than attempt a formal proof.
To understand what the function Weierstrass
constructed is, you may want to read up on continuous
and differentiable function
s. Very briefly a continuous function is one whose graph
does not jump : you could follow the graph
with a pencil without ever having to lift the pencil from the paper. For example, the function which is 0 for all x<0 and 1 for all x ≥0 is not continuous: at 0 it jumps.
Intuitively, differentiability is the property that a function's graph is "smooth". For example a sine wave is differentiable : It has no sharp corners. On the other hand a function like ψ defined above is not differentiable everywhere: it has kinks at the integers.
It should not be difficult for anyone to make drawings of continuous functions that are not differentiable at some points. It is not very difficult to think of functions that are nowhere differentiable and that are nowhere continuous (a discontinuous function cannot be differentiable). A continuous nowhere differentiable function is more of a challenge. Mathematicians before Weierstrass scratched their heads, thought a bit and said that surely, the niceness brought by continuity must grant you differentiability, at least in a few places.
Weierstrass' counterexample was a wake-up call. Beyond proving people wrong, it also showed that analysis must be done carefully. We owe the rigorous definitions of differentiabilty and continuity of functions of the real line to Weierstrass.