It is known pretty well that a measurable function on $[0,1]$ need not be equal almost everywhere to a continuous function. The standard example one uses for this, that is the indicator function of $[0,1/2]$, however, happens to be continuous almost everywhere. Hence, I would like to know of an example of a measurable function on $[0,1]$ that cannot be equal almost everywhere to an almost everywhere continuous function on $[0,1]$. Any help shall be immensely appreciated. Thank you all in advance.
2026-03-29 03:02:26.1774753346
On measurable functions and continuity
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