Uniform step approximation on a compact interval ($\Vert f-g{\Vert }_{\infty }\le Lh$)

A uniformly continuous real function on a closed bounded interval can be approximated uniformly by finite step functions. Given $\epsilon >0$, 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 $\epsilon$. If the function has a Lipschitz bound $L$, a mesh of maximum length $h$ gives the explicit error bound $\Vert f-g{\Vert }_{\infty }\le Lh$.