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From this post, we have for any measurable $f$ defined on closed and and bounded $I$ and $\epsilon>0$, there exists a step function $h$ defined on $I$ and a measurable set $F$ subset of $I$ such that $|h-f|<\epsilon$ and $m(I \setminus F) < \epsilon$.

I have one exercise from lecture as follows:

For any measurable $f$ defined on closed and bounded intercal $I$ and $\epsilon>0$, there exists a step function $h$ and a continuous function $g$ such that $|f-g|<\epsilon$ and $|f-h|<\epsilon$ except on a set of measure less than $\epsilon$.

I have no idea how to construct such a continuous $h$ that satisfies the requirement above.

Any hint would be appreciated.

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    continuous close to step: at each jump, make it continuously slant up/down in a tiny interval. You can use $2$ $\varepsilon/2$s. – 2017-02-07
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    So for any discontinuities, I define an open interval centered at discontinuites with length $\frac{\epsilon}{2}$? – 2017-02-07

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