Are 「ないで」(when used adverbially) and 「ずに」 absolutely interchangeable?


I have been browsing the related questions in this site that deal with ず, ない,
ずに and ないで and they are very informative. However, **I couldn't find a post
dealing with ずに and ないで (adverbially) specifically**. That's why I used the
categorical word "absolutely" in my question, I want to focus on those.

After researching, it is clear to me that both expressions mean the same, but
I still wonder whether there is a difference in nuance, formality or whatever
aspect that makes them different in the slightest. Are they always
interchangeable no matter the sentence they are used in? To put it the other
way around, is there any situation in which 「ないで」(adverbially) can't be
replaced by 「ずに」 or vice versa?

For instance,

> 朝ごはんを食べ **ないで** 、会社にでかけた。  
>  朝ごはんを食べ **ずに** 、会社にでかけた。

よろしくお願いします！




Here's an example case: Vないで(ください) can also be used as a negative
request/volitionally, where Vずに won't be able to be used as a drop-in
replacement. This is similar to the positive form Vてください。

Edit: I believe there is a difference in nuance between the two adverbial
forms, with ず sounding more a bit more classical/mature (although it is used
in many different contexts) and being related to せざる and older Japanese forms.
I couldn't find a dictionary source, but it's in this chiebukuro question:
<https://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q1157256921>




Here's an example case: Vないで(ください) can also be used as a negative
request/volitionally, where Vずに won't be able to be used as a drop-in
replacement. This is similar to the positive form Vてください。

Edit: I believe there is a difference in nuance between the two adverbial
forms, with ず sounding more a bit more classical/mature (although it is used
in many different contexts) and being related to せざる and older Japanese forms.
I couldn't find a dictionary source, but it's in this chiebukuro question:
<https://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q1157256921>

