I heard it many times and I could infer that it means "all the time" or "without pause", "constantly". (Am I right?) But when trying to look it up, none of the dictionaries define it. Is this a correct English phrase?
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By the second means per second or for each second (of time).
For given telephone base rates and whole-unit billing, your bill will be cheaper when charged by the second than when charged by the minute: Say you made a one minute and five second call. If you are charged by the minute, and each minute costs 60c, it would cost 120c. But if you are charged by the second, and each second costs 1c, it would cost only 65c.
Some other uses emphasise rapid change:
Cell phone service providers offer billing method known as by the second. You pay for the seconds used.