Is eating by a fasting person forgetfully on Thursday considered breaking the fast, and is the custom of fasting on Mondays and Thursdays considered a pre-existing intention that permits the continuation of the fast in the event of forgetting the day?

A fast whose continuity is not obligatory—such as a voluntary fast, for example—requires a specific intention for each day. A voluntary fast is valid if the intention is made after Fajr during the day, provided that the fasting person has not eaten or drunk anything before forming the intention. If one has eaten or drunk, it is not valid to intend to fast that day after having consumed something that invalidates the fast.