10. (20 points) Given
$$
\lim _{x \rightarrow 0} f(x)=f(0)=1, f(2 x)-f(x)=x^{2}
$$

for any real number $x$. Find the analytical expression of $f(x)$.
10. When $x \neq 0$,
$$
f\left(\frac{x}{2^{k-1}}\right)-f\left(\frac{x}{2^{k}}\right)=\frac{x^{2}}{2^{2 k}}(k=1,2, \cdots, n) \text {. }
$$

Adding these $n$ equations, we get
$$
f(x)-f\left(\frac{x}{2^{n}}\right)=x^{2} \cdot \frac{\frac{1}{4}\left[1-\left(\frac{1}{4}\right)^{n}\right]}{1-\frac{1}{4}} \text {. }
$$

Let $n \rightarrow \infty$, then $f\left(\frac{x}{2^{n}}\right)=f(0)=1$.
The right-hand side of the equation above $=\frac{x^{2}}{3}$.
Thus, $f(x)=1+\frac{x^{2}}{3}(x \neq 0)$.
Also, $f(0)=1$ satisfies the condition, therefore,
$$
f(x)=1+\frac{x^{2}}{3} .
$$