Putnam 1954 A5

Let $f(x)$ be a real-valued function defined for $0<x<1.$ If $$\lim_{x \to 0} f(x) =0 \; \text{and} \; f(x) - f \left( \frac{x}{2} \right) =o(x),$$ prove that $f(x) =o(x),$ where we use the O-notation.