ILL 1989 Problem 17

Let $a \in \mathbb{R}, 0 < a < 1,$ and $f$ a continuous function on $[0, 1]$ satisfying $f(0) = 0, f(1) = 1,$ and

$$f \left( \frac{x+y}{2} \right) = (1-a) f(x) + a f(y) \forall x,y \in [0,1] \text{ with } x \leq y.$$

Determine $f \left( \frac{1}{7} \right).$