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It seems that for verbs which imply an action over a period of time, it does not matter if they are in simple present or present continuous, it still means the same:
On that day, I will be keeping you informed
On that day, I will keep you informed
I will be maintaining its well-being.
I will maintain its well-being.

But,
On that day, I will be doing it
is different from
On that day, I will do it.

However these two seems similar,
On that day, I will be doing that the whole day
On that day, I will do that the whole day.

Why is that so?
Comments  
I think it is not so. These verbs do vary in their meanings depending on the tense used.

For example

He keeps on looking at us. - (means that he is looking for some period of time and then he is looking away then, again, at us)

He is keeping on looking at us. (Here it means that he is looking constantly without stopping)

All your examples can be interpretered the same way.
TicceHe keeps on looking at us. - (means that he is looking for some period of time and then he is looking away then, again, at us)
He is keeping on looking at us. (Here it means that he is looking constantly without stopping)
Thanks Ticce. But I am absolutely sure that "He is keeping on looking at us" is wrong to native ears. You can use "is keeping" like "is keeping the person safe" or "is keeping the book in his bag" but not in this case because of the gerund "looking".

I would use "He keeps looking at us" to mean "looking at us" continuously or intermittently. It's like saying that he "maintains" or is "maintaining" his current activity of "looking at us" at this moment.
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AnonymousBut I am absolutely sure that "He is keeping on looking at us" is wrong to native ears.
Don't be so sure))) I have checked it with a native speaker and it was acceptable to his ear.
English is my first language.
"He is keeping on looking at us" sounds weird to my ears. I advise you to seek a second opinion. I'm certain it's incorrect. English speakers don't use it in this way.
AnonymousEnglish is my first language. "He is keeping on looking at us" sounds weird to my ears. I advise you to seek a second opinion. I'm certain it's incorrect. English speakers don't use it in this way.
Even if it sounds weird it doesn't mean that this is wrong. However, I agree it is a bit unusual.

See for yourself. This is a passage from a book

"Look!" Mabel was suddenly excited. "He's looking straight at us ! Look, he's keeping on looking at us, Arlene. (By Booth Tarkington - he is an American novelist)

The guy I checked it with is from the UK and he said that it was unusual but still acceptable.
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TicceThe guy I checked it with is from the UK and he said that it was unusual but still acceptable.
Thanks for pointing that out. I believe I've used the wrong word, "incorrect" is too harsh. But as the British fellow has told you, it's unusual and we do not normally use this in speech but perhaps in literature where more artful language is used. The average native speaker seldom use such a syntax to express a continuous looking done by a third party. Instead we usually use "he keeps (on) looking at us". If you don't believe me search Google with "keeping on looking", you'll yield a very low count. Using "He's keeping looking at us" instead of "He keeps looking at us" is even bizarre.

"What's my duty?"
"You just keep me informed"
I highly doubt "keep" here means to be informed only at times and not continuously being informed.

Anyway, I can't see your point. "keep" in the context illustrated is an action done over a period of time similar in meaning to "maintain" and not the "keep" as in "keep the book". Thus I don't really even see a difference "He keeps on looking at us" and "He's keeping on looking at us". It doesn't need to be in its continuous form. It's not like the difference between "He swims" vs. "He's swimming" because if it were that, your explanation would suffice. If you can, enlighten me. I will appreciate it. Moreover, there are verbs which are normally not used in continuous or can be used in both simple and continuous form:

E.g. 1. "He wants his food" and not "He's wanting his food"
2. "I have known him for years" and not "I have been knowing him for years"
3. "I have lived there since 1999" OR "I have been living there since 1999"

And what about the verb "maintain" which I've mentioned?