4. Let $f(x)$ be a function defined on the set of real numbers $\mathbf{R}$, and satisfies the following relations:
$$
\begin{array}{l}
f(10+x)=f(10-x), \\
f(20-x)=-f(20+x) .
\end{array}
$$

Then $f(x)$ is ( ).
(A) an even function, and also a periodic function
(B) an even function, but not a periodic function
(C) an odd function, and also a periodic function
(D) an odd function, but not a periodic function
4. C.
$$
\begin{array}{l}
f(20-x)=f(10+(10-x)) \\
=f(10-(10-x))=f(x)=-f(20+x) \\
\Rightarrow f(40+x)=f(20+(20+x)) \\
\quad=-f(20+x)=f(x) .
\end{array}
$$

Thus, $f(x)$ is a periodic function.
$$
\begin{array}{l}
\text { Also, } f(-x)=f(40-x)=f(20+(20-x)) \\
=-f(20-(20-x))=-f(x),
\end{array}
$$

Therefore, $f(x)$ is an odd function.