7.111 Among the four functions listed below, the function $f(x)$ that satisfies the property
$$
f\left(\frac{x+y}{2}\right)=\frac{1}{2}[f(x)+f(y)]
$$

is
(A) $\lg x$.
(B) $\frac{1}{x}$.
(C) $3 x$.
(D) $3^{x}$.
(China Beijing High School Mathematics Competition, 1986)
[Solution] From the figure, it is intuitively visible: the y-coordinate of point $A$ is $f(x)$, the y-coordinate of point $B$ is $f(y)$, and the y-coordinate of point $M$ is $f\left(\frac{x+y}{2}\right)$.

Then $A, M, B$ should be three collinear points, and only a linear function can meet this requirement, so we exclude $(A)$, $(B)$, and $(D)$, and choose $(C)$.

Alternatively, this problem can also be solved directly by substitution and verification.
Therefore, the answer is $(C)$.