Series of functions flourished in the 18th century, but only in the following century did precise theorems make it possible to determine their properties, particularly their continuity and differentiability. This vastly expanded the range of mathematical functions beyond combinations of algebraic operations on elementary functions such as powers, roots, logarithms, exponentials and trigonometric functions. In particular, it produced many examples of functions that defy intuition.
When the sum is discontinuous
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We are familiar with the optical illusion, illustrated by Maurits Cornelis Escher, of a fountain whose water seems to flow endlessly, only to return… to its starting point.
Here is an example of a periodic function that is continuous at every irrational point but has a downward jump at every rational point (that is, its right-hand limit at each such point is strictly less than its left-hand limit). A periodic function that is always decreasing? So much for intuition…
Let E(x) denote the floor of the real number x, and let h (x) be the difference x – E(x). This function (called the fractional part and often denoted {x}) is periodic because E(x+1) = E(x) + 1; hence h (x+1) = x + 1 – (E(x) + 1) = h (x). Moreover, for every x, | h (x) | ≤ 1. Finally, h is continuous at every non-integer point but has a downward jump at every integer; for example, h (1) = 0, whereas h (x) tends to 1 as x approaches 1 from below.
