Define $f:[0,2]\rightarrow\mathbb{R}$ by setting $f(x)=1$ if $x\not=1$ and $f(1)=3$. Find a partition $D$ of $[0,2]$ for which $S_D-s_D<2^{-1000}$.
2026-03-30 22:57:28.1774911448
Finding Partition, Riemanns Integral
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Then partition $P$ will have only four points: $$ t_=0,\,\,t_1=1-\delta,\,\,t_2=1+\delta,\,\,t_3=2. $$ Clearly $$ U(f,P)-L(f,P)=3\cdot 2\delta-1\cdot 2\delta=4\delta. $$ Hence you need $$ 0<\delta<2^{1002}. $$