$F:[0,1]\times[0,1]\longrightarrow R$
$ f(x)= \begin{cases} 1, & \text{y<x} \\ 0, & \text{y $\geqslant$x} \end{cases} $
i have a problem choosing my p∈P and proving the statement any guidance would be great
2026-03-27 00:58:10.1774573090
prove that $∀ε>0∃p∈P(U(f,p)−L(f,p)<ε)$
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Let $P_N$ be a partition of $[0,1] \times [0,1]$ into a mesh of $N \times N$ equally sized squares (each has an area of $1/N^2)$. The difference $U(f,P_N)-L(f,P_N)$ then equals $1/N$ (why?). Taking $N>1/\varepsilon$ gets the job done.