Whether you view your response as continuous depends on measurement properties of both the numerator and the denominator. If the denominator is the number of days in the month (which I am assuming it is), AND if the numerator is NOT limited to just a few days per month (at any point in the range - that is, whether the few days are near zero or near 15 or near 30), then you could probably treat the response as continuous. However, if either the numerator or the denominator has a restricted range, then treating the response as continuous is probably not reasonable.
How to model the response is another question entirely. You don't indicate whether the relative frequency is near 0 or near 1. If near zero, then you could probably model the response as a count with distribution Poisson or negative binomial and treat log(number of days in the month) as an offset variable. If the relative frequency is closer to 1 and you have a limited response range, then Poisson and negative binomial distributions are probably not reasonable.
In order to advise to any greater degree, we need to know more about the measurement characteristics of your data.