Correct answer: (C) f is continuous almost everywhere.
Explanation
A function continuous except on a null set is measurable: its restriction to the continuity set is Borel measurable, and arbitrary changes on a null set preserve Lebesgue measurability. Thus **C**. A sign choice on a nonmeasurable set can have |f|≡1 and be bounded by continuous constants, refuting A,B. Measurability of individual level sets alone does not ensure that inverse images of intervals are measurable.
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