Question

不定式的进行式to be doing有什么用法,

Answer 1

在seem,appear,happen,think,consider,believe,say,report等动词后,有时用不定式的进行式to be doing,表示这些动词的动作发生的同时,不定式所表示的动作或状态也正在进行着或持续着：
I happened to be cooking when the telephone rang. 当电话铃响时,我恰巧在烧饭.
The boy pretended to be sleeping when his mother came in. 当他母亲进来时,那孩子假装在睡觉.