7・118 Suppose $f(x)$ is defined for all real numbers $x$; for all $x, f(x)>0$; for all $a$ and $b, f(a) f(b)=f(a+b)$. Which of the following statements are true?
I . $f(0)=1$
II . $f(-a)=1 / f(a)$, for all $a$
III. $f(a)=\sqrt[3]{f(3 a)}$, for all $a$
IV. $f(b)>f(a)$, if $b>a$
(A) Only III and IV.
(B) Only I , III and IV .
(C) Only I , II and $\mathrm{N}$.
(D) Only I , II and III.
(E) All are correct.
(26th American High School Mathematics Examination, 1975)
[Solution] In the given condition $f(a) f(b)=f(a+b)$, let $a=0$ to get $f(0) f(b)=f(b)$, or $f(0)=1$.

Also, let $b=-a$ to get
$$
f(a) f(-a)=f(0) \text {, or } f(-a)=1 / f(a) \text {; }
$$

And $f(a) f(a) f(a)=f(a) f(2 a)=f(3 a)$, or
$$
f(a)=\sqrt[3]{f(3 a)} \text {. }
$$

The function $f(x)=2^{-x}$ satisfies the given definition, but it does not satisfy condition IV. Therefore, the answer is $(D)$.