Please be aware that the question has been edited and it is significantly different to when I answered it
I had just finished my work and was starting to read a book.
Q. The answer is "was starting." I think the answer is wrong by itself. Since "start" is a transient verb, it can't be used in present continuous.
I would suggest that the answer is correct
transient; adjective; lasting for only a short time; temporary:
Although an action may only last for a short time it may be continuous whilst it is happening. Temporary & Continuous are not mutually exclusive. The action of walking continues until you stop walking. The action of starting a car continues until the car has started and you release the starter.
Starting the car, Starting the race, starting a war
Q. How can “start” used in present continuous? How can “start” be used in the Present Continuous Tense?
Present Continuous: form; We use am, are, is + the -ing form of the verb
I am starting the car.
However in this case I think you are confusing the Present Continuous with the Past Continuous
Past continuous: form; We use was/were + the -ing form of the verb.
"I had just finished my work and was starting to read a book".
Past continuous: uses; Events happening at a particular time in the past
We generally use the past continuous to talk about actions and states in progress (happening) around a particular time in the past. It can emphasise that the action or state continued for a period of time in the past:
Where was Donna last night? I’m not sure. I think she was visiting her family. (action)
or
I remember that night. You were wearing that red dress. (state)
Ref CED Past continuous
Ref CED present participle -ing
-ing: used to form the present participle of regular verbs:
present participle; a form of a verb that in English ends in -ing and comes after another verb to show continuous action. It is used to form the present continuous:
In the sentences "The children are watching television", "The weather is getting colder", and "I heard him singing", "watching", "getting", and "singing" are present participles.