4. Given the function $F(x)=\left(\frac{1}{2}+\frac{1}{2^{1}-1}\right) \cdot f(x)(x \neq 0)$ is an even function, and $f(x)$ is not identically zero, then $f(x)($ ).
A. is an odd function
B. is an even function
C. may be an odd function or an even function
D. is neither an odd function nor an even function
4. A
$$
\begin{aligned}
F(-x) & =\left(\frac{1}{2}+\frac{1}{2^{-x}-1}\right) \cdot f(-x) \\
& =\left(\frac{1}{2}+\frac{2^{x}}{1-2^{x}}\right) \cdot f(-x) \\
& =-\left(\frac{1}{2}+\frac{1}{2^{x}-1}\right) \cdot f(-x) .
\end{aligned}
$$

Since $F(x)$ is an even function, $F(-x)=F(x)$, thus $f(-x)=-f(x)$, $f(x)$ is an odd function, and since $f(x)$ is not identically zero, $f(x)$ is not an even function.