Is fasting the 9th, 10th, and 11th of Muharram an innovation (bid'ah), and what is the justification for its permissibility by some scholars?

One must be cautious not to go to extremes in labeling something as an innovation (bid'ah). Some scholars have mentioned that the fasting of Ashura has three forms: fasting the day before it and the day after it, or fasting the ninth and tenth, or fasting the tenth and eleventh, or singling out the tenth for fasting. Sheikh Ibn Uthaymeen mentioned that the most complete form is to fast a day before it and a day after it. It is reported in a hadith from Ibn Abbas: "Fast the day of Ashura, and differ therein from the Jews, by fasting a day before it or a day after it." And in another narration: "Fast a day before it and a day after it." Fasting the eleventh is an increase in goodness, and fasting in Muharram is legislated. Therefore, fasting the three days cannot be considered an innovation.