Question 57
Let $f:[0,1) \rightarrow \mathbb{R}$ be a function that satisfies the following condition: if
$$
x=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{n}}= . a_{1} a_{2} a_{3} \ldots
$$
is the decimal expansion of $x$ and there does not exist a positive integer $k$ such that $a_{n}=9$ for all $n \geq k$, then
$$
f(x)=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{2 n}} .
$$
Determine $f^{\prime}\left(\frac{1}{3}\right)$.
$$
x=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{n}}= . a_{1} a_{2} a_{3} \ldots
$$
is the decimal expansion of $x$ and there does not exist a positive integer $k$ such that $a_{n}=9$ for all $n \geq k$, then
$$
f(x)=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{2 n}} .
$$
Determine $f^{\prime}\left(\frac{1}{3}\right)$.