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    Given g:[0,2]\to\mathbb{R} with the rules
    g(x) = 2,\text{  }0\leq x<1
         g(x)= 1, \text{  }x=1
        g(x)   = 2,\text{  } 1<x\leq 2
    by the definition prove that g is integrated!

    #2
    Given f:[0,1]\to \mathbb{R} and f\geq0
    Let P_1=\{0,x,1\}\text{ and }P_2=\{0,x,y,1\}
    Prove that L(P_1,f)\leq L(P_2,f)\leq U(P_2,f)\leq U(P_1,f)
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    What have you tried for each question? #1 is just a routine check of Riemann integrability, and #2 follows from results about suprema and infima.
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    for #1 i got
    U(P_n,f)=L(P_nf)=4-\frac{2}{n}
    so, the infimum of upper squares and the supremum of lower squares must be not equal.

    for #2
    i have tried it and it holds if the function monotonic increasing. I still have no idea to solve if the function is generally f\geq 0
 
 
 
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Updated: April 13, 2011
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