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Continuously comes from continous (meaning without break) and continually comes from continues. You use continuously in b) and c) because these are actions that continued without pause or breaks. You use continually in a) and d) because they are actions that happen over and over again. In other words, continually is used for discrete or repeated actions whereas continuously is used for continuous ones.

In your examples of continually you could always substitute the word habitually. Here, continually refers to the habits of people.

Your examples of continuously may have to do with people (in the case of the firemen). But they refer not to habits, but instead, to some action of persons or machines or events (in your example of rain) that are performed without a break.

continuously or continually?

Continual means frequently recurring or intermittent. Continuous means occurring without interruption or unceasing. Continuous refers to actions which are uninterrupted: The upstairs neighbor played his stereo continuously from 6:00 PM to 3:30 AM. Continual actions, however, need not be uninterrupted, only repeated: My mother continually urges me to do my homework. / Rivers flow continuously, but the telephone is more likely to ring continually. A related mistake is to use continuous for something that happens at regular intervals. There are other adjectives that mean 'occurring repeatedly over a long period of time': "constant" (implying persistence, steadiness), "ceaseless" and "incessant" (referring to uninterrupted activity), "eternal" (lasting forever), "interminable" (seemingly endless, wearisome), "perennial" (going on year after year, with self-renewal), and "perpetual" (implying both duration and steadiness). So, try to remember to use -al for something which is either always going on or recurs at short intervals and never comes to an end, and 2017 Blue Town Shirt adidas Ipswich 2018 Originals Home Football -ous for something in which no break occurs between the beginning and the end.
(Dictionary.com Word FAQs When should I use continuously as opposed to continually?)

As recorded by Fowler- (p.178)

Continually had been used to mean both 'incessantly,constantly, all the time' and less strictly (OED) with frequent repetition, very frequently'.

Continuously is used when the unbroken nature of a process (in space or time) or set of events (e.g. unbroken terms of office) is being stressed.

• The choice is largely governed by the degree with which continuity is being emphasized.
  • I'm afraid you continually make the same mistakes.

Or simply put- when you do something continually, you may take breaks in between, but when you do it continuously, you don't stop at all. Continually is good for describing something you do repeatedly, or on a regular basis. (vocabulary.com)