MathLabs

Problem 5

Let ff be a real-valued function defined for all real numbers xx such that, for some positive constant aa, the equation f(x+a)=12+f(x)−f(x)2f(x+a)=\frac12+\sqrt{f(x)-f(x)^2} holds for all xx. (a) Prove that ff is periodic. (b) For a=1a=1, give an example of a non-constant function with the required property.
Step 4 of 5: Define a non-constant example for a = 1
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In plain words

The construction is a two-state clock: on one unit interval the value is 11, on the next it is 12\frac12, and the pattern repeats. The recurrence maps 11 to 12\frac12 and 12\frac12 to 11, so shifting by one interval performs exactly the required state swap.

f(x)={1,2n≤x<2n+1,12,2n+1≤x<2n+2,n∈Zf(x)=\begin{cases}1,&2n\le x<2n+1,\\[2pt]\frac12,&2n+1\le x<2n+2,\end{cases}\qquad n\in\mathbb Z
Detailed analysis

For a=1a=1, define f(x)=1f(x)=1 when 2n≤x<2n+12n\le x<2n+1 and f(x)=12f(x)=\frac12 when 2n+1≤x<2n+22n+1\le x<2n+2, for an integer nn. These half-open intervals partition the real line, so the definition is unambiguous and ff is non-constant. It is also visibly periodic with period 22.