Integrability of f on (c,d) from (a,b): Proof

[Kate2010]

Homework Statement

If a<c<d<b and f is integrable on (a,b), show that f is integrable on (c,d)

Homework Equations

The Attempt at a Solution

I know that f is integrable on (a,b) iff for all e>0 there exists step functions g and h such that g [tex]\leq[/tex] f1_(a,b) [tex]\leq[/tex] h and I(g-h) <e
( 1_(a,b) in the indicator function and I(g-h) is the integral of the step functions)

I feel like this should allow me to fairly easily show that f is also integrable on (c,d) but I just don't know how to start.

Do I need to consider partitions?

Thanks.

 

[jbunniii]

I assume you mean [itex]I(h-g)[/itex], not [itex]I(g-h)[/itex].

To show integrability on the interval [itex](c,d)[/itex], consider the functions [itex]g|_{(c,d)}[/itex] and [itex]h|_{(c,d)}[/itex], which are the restrictions of [itex]g[/itex] and [itex]h[/itex] to the interval [itex](c,d)[/itex]. Are the restrictions still step functions? Do they satisfy the desired inequality?

 

[Kate2010]

Yes I did sorry.
Thanks :) so if I use those functions that take the same value on (c,d) and are 0 elsewhere I think I can see how it goes.