148 The function $f(x)=\left|x^{2}-a\right|$ has a maximum value $M(a)$ in the interval $-1 \leqslant x \leqslant 1$. The minimum value of $M(a)$ is
$\qquad$
$148 \frac{1}{2}$. From the problem, we know
$$
\left\{\begin{array}{ll}
M(a)=1-a & a \leqslant \frac{1}{2}, \\
M(a)=a & a \geqslant \frac{1}{2} .
\end{array}\right.
$$

Therefore, when $a=\frac{1}{2}$, $M(a)$ is minimized, with the minimum value $M\left(\frac{1}{2}\right)=\frac{1}{2}$.