Question

Homework check! Can you please see if my proof is 1 rigorous enough, and 2 correct. Thanks!!!
Prove:
If \[f\in R[a,b]\] and \[[c,d]\subset [a,b]\] then the restriction \[f|_{[c,d]}\] is in R[c,d].

Answer 1

WLOG assume c<d. Then we have a<c<d<b. Since [c,d] is contained in [a,b]. by thm 5.2.2

Answer 2

We know that if \[f\in R[a,b]\] then for every e is an element of [a,b] \[\int_a^b f=\int_a^ef+\int_e^bf\]

Answer 3

which means f must be riemann integrable on [a,e] and [e,b]. So now we can write

Answer 4

\[\int_a^bf=\int_a^cf+\int_c^df+\int_d^bf\] by thm 5.2.2, this means f must be riemann int on [a,c] [c,d] and [d,b] in order for \[f\in R[a,b]\]** The restriction means our domain is [c,d] for f. Therefore by ** [c,d] is riemann int.

Answer 5

end pf

Answer 6

thm 5.2.2 is essentially what I stated with the a<e<b interval up top

Answer 7

i have no clue about restrictions @ikram002p @ChristopherToni

Answer 8

restriction just means evaluate f on whatever interval they restrict it to

Answer 9

ie the domain is only [c,d] here

Answer 10

@eliassaab , can you check?

Answer 11

@aum @zepdrix

Answer 12

@zzr0ck3r is online

Answer 13

what is R[a,b] and what is theorem 5. what ever

Answer 14

Riemann set essentially riemann integrable and the thm is in a second comment below

Answer 15

http://www.jirka.org/ra/realanal.pdf page157

Answer 16

looks fine.

Answer 17

does it need anything, is there anything that needs elaborated on?

Answer 18

needs some dx's :)

Answer 19

lol, yea, we don't use those in this class "in order to leave the variable of integration open"

Answer 20

prob a good idea:)

Answer 21

thanks, I keep getting nailed on points because my proofs are not rigorous enough.

Answer 22

I don't see what else you could add to it.

Answer 23

yea.. tell me about it, but with how this class is going I'll still get docked somewhere

Answer 24

thanks for looking at it :)

Answer 25

for sure!