Is my fast still valid?

Today I woke up at about 10:00. I remembered that it was a day of Ramadan, however I forgot whether or not I had made my intention to fast or not. Because of this, I ate food, thinking that it was allowed since I was unsure, and also thinking that if I had made the intention, that this would break the fast anyway. Thinking that my fast, if ever valid, was invalidated, I continued to eat again later. Afterward I realized that I had indeed made the intention to fast the night before (and also that morning, unknowing that making the intention after Fajr was allowed.) Should I keep this fast and make up for eating mistakenly, or should I stop fasting (eat again before sunset) and make Kaffara?