Many writers use continuous and continual as if they were exact synonyms, but my English teachers taught their students to distinguish between them.
Both adjectives describe duration.
Continuous indicates duration without interruption.
Ex. The continuous humming of the fluorescent lights gave him a headache.
Continual indicates duration that continues over a long period of time, but with intervals of interruption.
Ex. The continual street repair disrupted traffic for nearly two years.
The adverbs continuously and continually preserve the same distinction:
Ex. The child screamed continuously as long as its mother was on the telephone.
In this part of the country it rains continually during April.
Here are some quotations from newspapers that illustrate that usage:
… gym membership or a magazine subscription. Some will be direct debits and standing orders, but others will be “continuous payment authorities” (also called recurring payment authorities, recurring transactions or recurring payments … (www.theguardian.com)
… toward excellence. However, many firms from the smallest to the largest fail to develop and maintain an attitude of continuous improvement. … (www.chicagotribune.com)
… Uefa’s numerical record of events he almost fails to register, a ghost player draped in his own invisibility cloak of continual peripheral motion. Despite all of this, Milner has still managed to remain a largely uncontested select … (www.theguardian.com)
… modern art found a way of domesticating Abraham’s powerful idol-smashing instincts by allowing itself to be continually driven by the iconoclastic urge. Or rather, to be driven by iconoclasm simply for its own sake. It is … (www.theguardian.com)