Question

Real Analysis: Define f: [0,1] --> $\Bbb R$ by f(x) = {0, x $\in$ $\Bbb Q$$\cap$ [0,1] ; 1, x $\in$ [0,1]\$\Bbb Q$ }

(a) Identify U(f) = inf{U(f, P): P $\in$ $\script P$(a,b)}

(b) Prove or disprove that f is Darboux Integrable.

Thanks in advance!

Answer 1

fo= So la & Xtan [0, 1] xeon [0,1] P[0, 2] := set g all partition g [of] Now we are going to find copper & laves integral og fix). het pke any partion f '10.2 ] destined of xo=0< x, 4X₂ <X3 <... (x2=1 Now we find copper integral for his partiotion . ie. moi sup tex) le. supremum of tw) in [xi-u*;] is mi " xe Exiuxi] s mi ax; = m; ax + m₂ 4X2 + ... + Mn Olm NOC Now we can see costly for meg partion p i'n each subinreal Exit.] t' there is both national of irrational No. so mi = 10 tit{1,2, . . n.

miar; = = 1 (x,-Xo) + 1. 18 - x) + - - -+ 1. (Antal (xn--X) far encey rational parention & irrational Now the abone logic ir se i.e. so subintonds contain's Both & PE P[0, 1] . Ulf,P) = 1 [tax = spinth U (+,P) / PEP [1]} This is darbst uphorsom. firsan a Now we evaluate it's lower Integral, 214,p) - m; ax; for my partitom p max, + m₂ L X2t - it ma olm Now Notice that mi-o ti (By abane logic) » LI dIP) = 0 & PEP [0, 1] strah - sup. } 218,P) I PEPIJ} = 0 . - set it fention is not oburbox integrable.