7. Let $a, b \in \mathrm{R}$, then the minimum value of $\max \{|a+b|,|a-b|,|1-b|\}$ is

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【Analysis】Answer: $\frac{1}{2}$.
$$
\max \{|a+b|,|a-b|,|1-b|\}=\max \{|a|+|b|,|1-b|\} \geq \frac{|a|+|b|+|1-b|}{2} \geq \frac{|a|+1}{2} \geq \frac{1}{2} \text {. }
$$

Equality holds if and only if $a=0, b=\frac{1}{2}$.