In 1966 Karl Kießwetter [1] introduced a simple example of a continuous nowhere differentiable function. He claims that the example can be understood by talented high school seniors.

We illustrate Kießwetter's function in two different ways: (1) as a limit of partial sums ("series"), and (2) as an intersection of sets ("fractal"). The second definition is due to Gerald A. Edgar [2].

The first example of a continuous nowhere differentiable function was found by Karl Weierstraß in 1872.

More examples were found by Bernard Bolzano, Charles Cellérier, Bartel L. van der Waerden, Teiji Tagaki, Theophil H. Hildebrandt, and others.

References

[1] K. Kiesswetter, "Ein einfaches Beispiel für eine Funktion, welche überall stetig und nicht differenzierbar ist," Mathematisch-Physikalische Semesterberichte, 13, 1966 pp. 216–221.

[2] G. A. Edgar, Measure, Topology, and Fractal Geometry, New York: Springer-Verlag, 1990.