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Let \(f(x) = x\) on \([0, 1]\). For \(n \in \mathbb{N}^*\), define the step function \(\varphi_n\) by \(\varphi_n(x) = k/n\) on \([k/n, (k+1)/n[\) for \(k = 0, \dots, n-1\), and \(\varphi_n(1) = 1\). Compute \(\|f - \varphi_n\|_\infty\).
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