3. The function $f(x)=\frac{x}{1-2 x}-\frac{x}{2}$
(A) is an even function but not an odd function
(B) is an odd function but not an even function
(C) is both an even function and an odd function
(D) is neither an even function nor an odd function
(A)
3. 【Analysis and Solution】The domain of the function $f(x)$ is $(-\infty, 0) \cup(0,+\infty)$, when $x \neq 0$, because
$$
f(-x)=\frac{-x}{1-2^{-x}}-\frac{-x}{2}=\frac{-x 2 x}{2^{x}-1}+\frac{x}{2}=\frac{x+x\left(2^{x}-1\right)}{1-2^{x}}+\frac{x}{2}=\frac{x}{1-2^{x}}-x+\frac{x}{2}=\frac{x}{1-2^{x}}-\frac{x}{2}=f
$$
$(x)$, so $f(x)$ is an even function. It is obvious that $f(x)$ is not an odd function, hence the answer is A.