Proving Continuity of Compositions: Sets of Measure Zero

[quantarb]

Homework Statement

I am trying to prove that if f is continuous almost everywhere on [a,b], and if g is cont a.e. on [c,d], with
f[a,b] contained in [c,d], then g composite f is cont. a.e.

The Attempt at a Solution

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Originally, my proof went something like this:

f is cont. a.e. implies f is Riemann integrable

g is cont. a.e. implies g is Riemann integrable

since f and g are Riemann integrable, g composite f is Riemann integrable (*)

A Riemann integrable function is cont. a.e., thus g composite f is cont. a.e.

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I was satisfied with this until I realized (*) might not be true, as I couldn't prove it as a Lemma.

Also, I am a newby to this forum - does anyone know if there is a way to LaTex my input?

Thanks

 

[Eynstone]

The points of discontinuity of g composite f are contained in the union of those of f & g.
Since the union of two null sets is also null, gof will be continuous a.e.