= Uniform step approximation on a compact interval
{title2=$\|f-g\|_\infty\le Lh$}
A <uniformly continuous> real <function> on a closed bounded interval can be approximated uniformly by finite <step functions>. Given $\varepsilon>0$, choose a mesh smaller than the uniform-continuity distance and set the approximant equal to a sampled value on each interval. Then the <supremum norm> of the approximation error is at most $\varepsilon$. If the function has a <Lipschitz bound> $L$, a mesh of maximum length $h$ gives the explicit error bound $\|f-g\|_\infty\le Lh$.
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