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Is this true or wrong? How to prove it ？

Qing Huan
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1 Answers1

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It is true.

1) One of the definitions for a continuous function is that the pre image of an open set is open.

2) A function is measurable iff $f^{-1}((-\infty, t))$ is measurable.

In our case, $f^{-1}((-\infty, t))$ is open and every open set is measurable, thus, it's measurable.

I assumed you know 1 and 2, if not I can clarify.

Shaked
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    Here the O.P. is asking whether an almost everywhere continuous is measurable. Your proof only works if $f$ is continuous everywhere. – Luigi M Sep 16 '17 at 16:15