$f$ is a real valued function defined for all real numbers $x$ such that $0\le f(x)\le \dfrac12$ and for some fixed $a>0$, $f(x+a)=\dfrac12-\sqrt{f(x)-(f(x))^2}$ for all $x$. Show that the function is periodic.
My attempt:
I could not really make an attempt, I tried testing $a$ as the period of the function but that didn't really help. My next attempt was considering an arbitrary period and then bring it under the definition. But, that didn't help either. Please help. Thank you.
Best Answer
What about squaring both sides?
$$\left[f(x+a) - \frac 12 \right]^2= f(x) - f(x)^2 = \frac 14 - \left[ f(x) - \frac 12 \right]^2.$$
It follows that $$\left[f(x+2a) - \frac 12 \right]^2= \frac 14 - \left[ f(x+a) - \frac 12 \right]^2 = \left[ f(x) - \frac 12\right]^2.$$
Consequently $$ \left| f(x+2a) - \frac 12 \right| = \left| f(x) - \frac 12 \right|$$ but since $0 \le f \le \dfrac 12$ both arguments are nonpositive and you get $f(x+2a) = f(x)$.