Show that the only function $f$ satisfying this property is the constant function $f(x) = a$ 
Let $f : (−1, 1) \rightarrow R $ be a continuous function with the property that
  $f(x) = f(x^4)$ for all $x$ and $f(\frac{1}{2}) = a$. Show that the only function $f$
  satisfying this property is the constant function $f(x) = a$.


We have $a=f(1/2)=f(1/2^4)=f(1/2^{16})$ and so on. So by continuity $f(0)=a$,  
Suppose that $f(k)=b$ for some $k$ with $|k|\lt 1$. The same argument shows that $f(0)=b$.
