Is the fast of one who began the day fasting, but did not intend to fast, valid? And is the fast of one who woke up in a state of major ritual impurity (junub) in a month other than Ramadan valid?

If a woman observes an obligatory fast, such as in Ramadan (definition), without having formed the intention before Fajr, her fast is invalid according to the majority of scholars, and she must make it up. However, Hanafis permit delaying the intention until Duha (late morning). As for supererogatory fasts, she may form the intention to fast after morning breaks, provided she has not done anything that breaks the fast. If she wakes up in a state of major ritual impurity (junub) during Ramadan or otherwise, her fast is valid, and there is nothing against her.