Compare Didache 8:1 and Mishnah Taanit 2:9

“... The [...] in Did 8:1 probably is a general reference to (pious) Jews. During a period of national calamity, especially in times of drought affecting the agrarian society, the religious leaders requested a public fast. There may have been one or more regular fasts in addition to Yom Kippur or the Day of Atonement in the first century CE, but in time of need, such as the shortage of rain in the autumn, special fast-days were added. Mondays and Thursdays were market days, and because the villagers used to gather in town on these days, they offered a good opportunity for Scripture reading and tribunal sessions. Fasts were connected with the public assemblies and this custom was almost certainly in existence before 70 CE. [...] Besides public fasts that were obligatory for all, there were also individual private days of fasting. In Luke 18:12, a Pharisee fasts twice a week, but it is not clear whether the days referred to are Monday and Thursday. From evidence in other sources, however, one may draw the conclusion that, also for private fasts, Mondays and Thursdays were the favoured days. ...”