4. Given $M=\left\{(x, y) \mid y \geqslant x^{2}\right\}, N=\left\{(x, y) \mid x^{2}+(y-a)^{2} \leqslant 1\right\}$, then the necessary and sufficient condition for $M \bigcap N=N$ to hold is
A. $a \geqslant 1 \frac{1}{4}$
B. $a=1 \frac{1}{4}$
C. $a \geqslant 1$
D. $0<a<1$
Solve:
Given the system $\left\{\begin{array}{l}y=x^{2}, \\ x^{2}+(y-a)^{2}=1\end{array}\right.$
$$
\Rightarrow y^{2}-(2 a-1) y+a^{2}-1=0 \text {, }
$$

When $\Delta=(2 a-1)^{2}-4\left(a^{2}-1\right)=0 \Rightarrow a=\frac{5}{4}$, the circle is tangent to the parabola. From the graph, we know $a \in\left[\frac{5}{4},+\infty\right)$. Therefore, the answer is $A$.