Example 7 (2005 Fujian Province College Entrance Examination Question) $f(x)$ is a function defined on $\mathbf{R}$ with a period of 3 and is an odd function, and $f(2)=0$. Then the minimum number of solutions to the equation $f(x)=0$ in the interval $(0,6)$ is
A. 2
B. 3
C. 4
D. 7
Solve: From $f(-x)=-f(x), f(x+3)=f(x)$, we know that the graph of $f(x)$ is symmetric about the point $\left(\frac{3}{2}, 0\right)$, i.e., $\rho\left(\frac{3}{2}\right)=0$. Also, $f(2)=0$. By symmetry, $f(1)=0$. And $f(3)=f(0)=0$, so within one period segment $(0,3]$, the solutions to the equation $f(x)=0$ are four: $x=1, x=\frac{3}{2}, x=2, x=3$. Thus, by a period of 3, in the interval (3,6), the solutions to the equation $f(x)=0$ are 3: $x=4, x=\frac{9}{2}, x=5$. Therefore, the minimum number of solutions to the equation in $(0,6)$ is 7. Hence, the answer is D.

Think: Does a function $f(x)$ that meets the conditions listed in the example exist? It needs to satisfy:
(1) The domain of $f(x)$ is $\mathbf{R}$.
(2) $f(x)$ is a periodic function with a period of 3.
(3) $f(x)$ is an odd function over its domain.
(4) $f(2)=0$.
(5) The equation $f(x)=0$ has exactly 7 solutions in the interval $(0,6)$.

Solution (using the construction method): Construct the function
$$
f(x)=A \sin \pi x \cos \frac{\pi}{3} x \text{ (where } A \text{ is a constant, } x \in \mathbf{R} \text{). }
$$

It is easy to verify:
(1) The domain of $f(x)$ is $\mathbf{R}$.
(2) $f(x)$ is periodic with a period of 3 in its domain.

Because $f(x+3)=A \sin \pi(x+3) \cos \frac{\pi}{3}(x+3)$
$$
\begin{array}{l}
=A \sin (3 \pi+\pi x) \cos \left(\pi+\frac{\pi}{3} x\right) \\
=A(-\sin \pi x)\left(-\cos \frac{\pi}{3} x\right) \\
=A \sin \pi x \cos \frac{\pi}{3} x \\
=f(x),
\end{array}
$$

So $f(x)$ is periodic with a period of 3 in its domain.
(3) $f(x)$ is an odd function.

Because $f(-x)=A \sin (-\pi x) \cos \left(-\frac{\pi}{3} x\right)=A(-\sin \pi x) \cos \frac{\pi}{3} x=-f(x)$, so $f(x)$ is an odd function.
(4) $f(2)=0$.
(5) $f(x)=0$ has 7 solutions in the interval $(0,6)$, which are $x=1,1.5,2,3,4,4.5,5$.
At this point, we can say that the minimum number of solutions to $f(x)=0$ in the interval $(0,6)$ is 7, and its graph is shown in Figure 4-1-4.
This provides a specific example of the abstract function described in Example 7.
As shown in Figure 4-1-6, when $a>0$, and $a \leqslant k^{\prime}$ (where $k^{\prime}$ is the slope of the line passing through the origin $O$ and point $A(2 k+1,1)$), the graphs of the functions $y=u x, y=(x-2 k)^{2}$ have two different intersection points in the interval $(2 k-1,2 k+1]$.
Figure 4-1-6
Because the equation of the line $O A$ is $y=\frac{1}{2 k+1} x$. So $k^{\prime}=\frac{1}{2 k+1}$. Hence $0<a \leqslant \frac{1}{2 k+1}$,
i.e., the set sought is
$$
M_{k}=\left\{a \left\lvert\, 0<a \leqslant \frac{1}{2 k+1}\right.\right\} .
$$

Note: In the problem, there are relatively unfamiliar and abstract terms and symbols. Such problems often have a mathematical background. The textbook does not provide ready-made concepts or symbols, so understanding these unfamiliar and abstract terms and symbols is key to solving the problem.

When focusing on key phrases and sentences, if the involved terms and symbols are relatively abstract, such as in this problem, making them concrete is an effective strategy. A common approach is to start with special values or use intuitive graphs to represent them.

Math problems often require changing the form of concepts, simplifying the intermediate steps from conditions to conclusions, decomposing the connections between the conditions of the proposition, and hiding the mathematical ideas and background involved in the problem. Therefore, when solving the problem, it is necessary to uncover these essences and patterns through the text and carefully deliberate on the key phrases and sentences.