The difference between As a rule and Regularly
When used as adverbs, as a rule means in general, whereas regularly means with constant frequency or pattern.
check bellow for the other definitions of As a rule and Regularly
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As a rule as an adverb (idiomatic):
In general; most often.
Examples:
"We go fishing every Sunday as a rule, but today the lake is frozen."
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Regularly as an adverb:
With constant frequency or pattern.
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Regularly as an adverb:
normally; ordinarily.