As far as I know, REMEMBER is a state verb and can't be used in continuous form.
It's not that they can't be used in continuous form. Refer to the following explanation which is more helpful:
"Some verbs [believe, doubt, think, remember, ...] are not usually used in the continuous form, even when we
are talking about temporary situations or states. These are called
Stative verbs. Emphasis added.
It is unlikely that there is any grammatical problem with using the continuous form, but if you do so inappropriately it may feel unnatural:
** Are you belieiving in God, Father?
This should likely be recast as:
Do you believe in God, Father?
However, sometimes using unnatural sounding language is done on purpose and may be appropriate. For example, in 2003 McDonald's chose this slogan:
I'm lovin' it.
To say I'm loving something is unnatural for the same reason that saying are you belieiving in God is unnatural. Perhaps the fast food chain chose it because the unnatural sound makes it stand out. Grammar Girl discusses this phrase specifically:
Grammar Girl - Is "I'm Loving It" Proper Grammar?
-ing not necessarily a continuous action
However, your -ing examples are not exactly to express continuous actions:
- He started remembering his past days and got really emotional about it.
In 1. you are using a verb in combination with another verb: (Subject) (verb 1) (verb 2). Examples:
Bob enjoys cooking.
Alice stops working at five.
I want to believe.
I'm writing to ask for your advice.
Visit the airline's Web site to book a flight.
In this sort of formulation, should the second verb in the pattern use the -ing form (cooking, working), or the to+infinitive form (to ask, to book, to arrange, to believe)? Which one to choose seems to depend on the verb. This is discussed in the following article:
Cambridge Dictionary - Verb patterns: verb + infinitive or verb + -ing?
However, I would treat the advice in the Cambridge article as guidelines, not as grammatical rules. Cambridge's list of verbs also seems quite incomplete to me. And some verbs in their "-ing" list or their "-to+infinitive" list seem like they could go either way to me:
I don't like to cook in the evening.
Although the article would suggest that you should recast as
I don't like cooking in the evening.
Is this better? Maybe. But I don't think there's anything wrong with saying I don't like to cook, either. You may also find cases where Cambridge's suggested form (e.g., cooking) may yield examples that sound quite wrong:
** I want cooking.
Do not say this. Instead, say I want to cook.
Seeing is believing
- Believing in god is very important.
In this example, your -ing form is used to make a verb (believe) into a noun phrase. You can see this by substitution:
- [Belief] in god is very important.
Some verbs have natural noun counterparts (believe and belief, see and sight, ...), but most do not. In general a verb can be made into a noun phrase by using the -ing form or by using the to+infinitive form. Doing so effectively 'nounifies' a verb:
Seeing is belieiving.
Cooking for beginners.
To go the extra mile.
To boldly go where no man has gone before.
To Kill a Mockingbird.
To decide which form to choose, you might consult the Cambridge article again for guidelines, but in many cases the choice may be a matter of taste. For example, Harper Lee could have decided to title her 1960 novel Killing a Mockingbird instead, and the famous Star Trek phrase could have been rendered as Boldly going where no man has gone before. Those alternatives would have changed how the phrases feel, but grammatically I don't see any problem. For example, if I saw a book titled Killing a Mockingbird, I would not think it is ungrammatical. Killing in this case seems to feel more brutal and violent, whereas to kill seems to feel a bit more neutral and clear-headed.
To see is to believe.
** To cook for beginners.
Going the extra mile.
To see and going sound fine, but choosing to cook for the last example sounds odd, perhaps because it is likely to be confusing: Is it a book about cooking where you cook for beginners? Cooking for beginners might also be misinterpreted in that way, but using cooking seems to make the intended meaning clearer: It is a book about cooking and the book is for beginners.
Cooking in the evening is great.
To cook in the evening is great.
In this case, confusion seems very unlikely, so either formulation seems plausible to me. Although cooking may be preferred, if we follow the guidelines from the Cambrige article. Finally, the position of the word or the phrase used may also dictate which form you choose:
A beginner's guide to cooking.
Probably because the word to already appears directly before the action, it is very unlikely that you would choose the to+infinitive form in this case. This would the case no matter what you tried to substitute for cooking (agreeing, arranging, seeing, eating, ...).