# Lusin's theorem and continuous function

Lusin's Theorem: Let $f$ be a real-valued measurable function on $E$. Then for each $\epsilon > 0$, there is a continuous function $g$ on $\mathbb{R}$ and a closed set $F$ contained in $E$ for which $f=g$ on $F$ and $m(E \setminus F) < \epsilon$.

Now in the exercises: For the function $f$ and the set $F$ in the statement of Lusin's theorem, show that the restriction of $f$ to $F$ is a continuous function.

In Royden's proof, on p. 67 he mentions that $f_n=g_n$ and $f_n \to f$ uniformly on $F$. He goes on to say "the uniform limit of a continuous functions is continuous, so the restriction of $f$ to $F$ is continuous on $F$." Is this question really asking for me to just restate a line from his proof?

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It seems to me that it is asking for a restatement of part of the theorem itself? If $g$ is continuous on $\mathbb{R}$, and $f = g$ on $F$, then $f$ is continuous on $F$, right? –  Dan Brumleve Nov 5 '12 at 7:52
If everything's exactly as you've reproduced it, I think there's probably a mistake in the question-if this is homework, I'd bring it up with your professor. There are also errata for Royden available via Google. –  Kevin Carlson Nov 5 '12 at 8:00
Question: imgur.com/m0nHj. Theorem: i.imgur.com/NdUvo.jpg. –  abet Nov 5 '12 at 8:07