Практическая грамматика

1. Physical or mental ability.
The little boy could not open the heavy door.
He started to run as fast as he could.
I can study by myself (London)
Can is often used with verbs that concern the mind and the senses, such as believe, see, here, feel.
In the meaning of ability it can be used only with the Indefinite Infinitive.
2. Possibility.
a) due to circumstances:
I could not go to the theatre yesterday because I was busy.
From where he sat he could see a cluster of apple trees
in blossom. (Galsworthy)
I can bathe when it’s warmer. (Maurier)
You can have the book when I have finished it.
Can I have my photo taken?
In the last two sentences can comes close to may expressing permission, but it does not so much indicate possibility depending upon the will of the person addressed (which is the essence of permission) as possibility depending on circumstances.
b) due to the existing laws:
The more she studied, the less sure she became, till idly turning the pages, she came to Scotland. People could be married there without any of this nonsense. (Galsworthy)
You cannot play football in the street (i. e., you have no right).
Can expressing possibility is used with the Indefinite Infinitive only.
Note. Could is often used in polite requests.
Could you give us a consultation to-morrow?
3. Incredulity, doubt, astonishment (they are closely connected). In these meanings can is used with all the forms of the Infinitive in interrogative and negative sentences, though astonishment is expressed only in interrogative sentences.
«That is not true!» exclaimed Linton, rising in agitation. «It cannot be; it is incredible, it cannot be.» (Ch. Bronte)
«... Не может этого быть, это невероятно, не может
этого быть!»