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Integral Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to...

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Advance Math

Integral

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

Information:

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

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

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

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