4. Given $M=\left\{(x, y)\left|y \geqslant x^{2}\right|, N=\left\{(x, y) \mid x^{2}+(y-a)^{2} \leqslant 1\right\}\right.$. Then, the necessary and sufficient condition for $M \cap N=N$ to hold is:
(A) $a \geqslant 1 \frac{1}{4}$
(B) $a=1 \frac{1}{4}$
(C) $a \geqslant 1$
( D ) $0<a<1$
4.【Analysis and Solution】As shown in the figure, $M$ represents the region inside the parabola $y=x^{2}$ including the boundary, and $N$ represents the circular region with center at $(0, a)$ and radius 1 (including the circumference). Its equation is $x^{2}+(y-a)^{2}=1$.
The necessary and sufficient condition for $M \cap N=N$ to hold is $M \subseteq N$. That is, the circular region $N$ is inside the opening of the parabola $y=x^{2}$, and can have common points with the boundary curve $y=x^{2}$.
Let $(x, y)$ be any point on the parabola $y=x^{2}$. From the distance relationship between it and the center $(0, a)$, the required necessary and sufficient condition is
$$
\left\{\begin{array}{l}
x^{2}+(y-a)^{2} \geqslant 1 \\
y=x^{2} \\
a \geqslant 1
\end{array} \Leftrightarrow \left\{\begin{array}{l}
y^{2}-(2a-1)y+a^{2}-1 \geqslant 0 \\
y=x^{2} \\
a \geqslant 1
\end{array} .\right.\right.
$$

For any $y \in \mathbf{R}$,
$y^{2}+(1-2a)y+(a-1) \geqslant 0$ always holds.
$$
\therefore \Delta=(2a-1)^{2}-4\left(a^{2}-1\right) \leqslant 0 \text {. }
$$

Solving gives $a \geqslant \frac{5}{4}$.
$\therefore$ The necessary and sufficient condition for $M \cap N=N$ is $a \geqslant 1 \frac{1}{4}$. Choose (A).
In fact, the correct answer (A) can also be determined by intuitive elimination. From the figure, $N$ is the circumference and interior of a circle with center $(0, a)$ and radius 1. When $a$ is a sufficiently large positive number, it must be that $M \cap N=N$. Thus, options (B) and (D) are eliminated.

When $a=1$, it is easy to find that the parabola $y=x^{2}$ and the circle $x^{2}+(y-a)^{2}=1$ have three intersection points $(0,0),(1,1)$, and $(-1,1)$, so $N \not \subset M$, eliminating (C).

Since in each sub-question, among the four options (A), (B), (C), and (D), "there is one and only one correct answer", the correct answer must be (A).