5. (2007 Anhui Province College Entrance Examination Question) The function $f(x)$ defined on $\mathbf{R}$ is both an odd function and a periodic function, with $T$ being one of its positive periods. If the number of roots of the equation $f(x)=0$ in the closed interval $[-T, T]$ is denoted as $n$, then $n$ could be
A. 0
B. 1
C. 3
D. 5
5. D Since $f(x)$ is an odd function on $\mathbf{R}$, for $x \in \mathbf{R}, f(-x)=-f(x)$. Let $x=0$. We get $f(0)=-f(0)$, hence $f(0)=0$. Also, $f(x)$ has a period of $T$, so $f(-T)=f(T)=f(0)=0$.
$f(x)$ has at least one more root in $[0, T]$, otherwise $f(x)$ would be positive (or negative) throughout $[0, T]$. By the property of odd functions, $f(x)$ would be negative (or positive) throughout $[-T, 0]$, which contradicts the periodicity of $f(x)$ with period $T$. Choose D.