Question

Integral: If you know all about it you should be easy to prove.....

Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g).

Information:

g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0

g is discontinuous at every rational number in[0,1].

g is Riemann integrable on [0,1] based on the fact that Suppose h:[a,b]→R is continuous everywhere except at a countable number of points in [a,b]. Then h is Riemann integrable on[a,b].

f : [0,1]→R defined by (f(x) =0 if x = 0) and (f(x)=1 if 0 < x≤1)

f is integrable on [0,1]

Answer 1

We know that if f and g are both Riemann Integrable on [a,b] and f<=g on [a,b] then integration of f on [a,b] is <= integration of g on [a,b]. Here given that both f and g are Integrable over [a,b] and f <= g on [a,b]. Also I have written the property of a function if it Riemann Integrable and the definition of L(f), L(g), U(f), U(g). So it's very easy to prove. If you have any doubt then please leave a comment. I'll try my level best to solve your doubt.
Given that, f g both are Riemann Integrable. fug on [ay] Now, : f, g E R [a,b] re. both Riemann intogable on [9,67 bĪ a St=st = st . and so von, L. (B) = & Juf L (8)= 8 g a I ug on . [a, b] * 54 559 pass by O KO & L(I) < L (9)

Again, u(t) = at UCO) = 5 fag on [a,b] V - Stvo by 0 4 a a U (7) SV (g) on [a,b]
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