In the radiation example of the introduction to this Module, we explained why, after a whole number of weeks, the radiation level from a sample of material with a half-life of one week would be given by where is the initial radiation level. But we don’t expect the radiation to drop suddenly at the end of each week. Rather it falls gradually, and for example, after each day should be reduced just a little – ie multiplied by a factor just a bit less than one. In fact, multiplying seven times by the daily reduction factor should give the weekly reduction by half. So the daily factor would be . Thus, after one day, i.e. at , the radiation level would be given by . So the equation applies also at .
Similar reasoning works for any rational number of weeks.
But the time could in principle be any real number and might not even be rational.
So we need to consider as a function of t for all real t.