The Notorious Jumping Function of Continuum County

Give an example of monotonic function on [0,1] which is discontinuous at all rationals.

Need a hint? Click here.

Click here for the complete solution.

Hint 1

Sneak up on it a few discontinuities at a time.

Hint 2

We can enumerate the rationals in the interval [0,1] in groups by their denominators as

and so on. For each group, invent a monotonic function that is discontinuous at each point of the group. Then what?

Hint 3

Step functions are the simplest monotonic functions with discontinuities; let's keep it simple. We can make a function discontinuous on the nth group by

Here are graphs of f_2, f_3, and f_4 :

Now how can we leverage these simple functions to get one function that is discontinuous at all rationals?

Hint 4

One way to combine functions is to add them together; we can't quite do that here because the sum won't converge, but we can weight the functions and use

Prove this is well-defined, monotonic, and discontinuous at each rational.

The Rest of the Solution

The sum function is:

• Well-defined: We have 0 \le f_n(x) \le 1 and therefore the sum converges by comparison with the geometric series \sum_{n=1}^\infty \frac{1}{2^n}
• Monotonic: if a < b then
  f(a) = \sum_{n=1}^\infty \frac{f_n(a)}{2^n} \le \sum_{n=1}^\infty \frac{f_n(b)}{2^n} =f(b)
• Discontinuous at each rational: if y > \frac{k}{m} then
  f(y) - f\left( \frac{k}{m} \right) = \sum_{n=1}^\infty \frac{f_n(y) - f_n(k/m)}{2^n} \ge \frac{f_m(y) - f_m(k/m)}{2^m} \ge \frac{1}{2^m} \; \frac{1}{m}

An Everywhere-Continuous, Nowhere-Differentiable Function

We know that if a function has a derivative at a point, then it is continuous at that point. The converse is not true: The absolute value function is continuous at 0, but does not have a derivative at 0, because it has a "corner" at 0. Can there be a function that is continuous at all points but differentiable at no point? Yes, there can! We can exploit the same idea we used in the original problem to define a continuous function that is not differentiable anywhere. The first such function was invented by Karl Weierstrass around 1861, and the following example was invented by John McCarthy in 1953.

Define

and periodically outside this interval.

Then define

See if you can discover how to prove it is nowhere differentiable.

References

• John McCarthy, "An Everywhere Continuous Nowhere Differentiable Function," American Mathematical Monthly, vol. 60 (1953), p. 709. Reprinted in A Century of Calculus, Part I, ed. Tom M. Apostol et al., Mathematical Association of America, 1969, p.156. Proof that the example given above is nowhere differentiable. Also available on the web here.
• E. C. Titchmarsh, The Theory of Functions, 2nd edition, Oxford University Press, 1939. Section 11.22 gives Weierstrass's original example of a nowhere differentiable function, and Section 11.23 gives a third example due to van der Waerden.
• Click here to view the original problem submitted by Deb.