Every bounded-variation function is uniformly approximable by step functions

Fix $\eta >0$. Let $A$ be the set of $x\in [a,b]$ for which there is a finite chain $a=x_0<\cdots <x_m=x$ such that the oscillation of $f$ on every open interval $(x_{j-1},x_j)$ is below $\eta$. The set contains $a$ and is bounded above by $b$, so $c:=\sup A$ exists by [L2].

Suppose $c<b$. The left limit at $c$ and the right limit at $c$ supplied by [L1] give one-sided intervals on which the oscillation is below $\eta$. Choose $x\in A$ in the left interval (use $x=a$ if $c=a$), append $c$ to its chain if necessary, and then append a point $y>c$ in the right interval. This puts $y$ in $A$, contradicting that $c$ is an upper bound. Hence $c=b$. The left limit at $b$ now lets a chain ending sufficiently near $b$ be extended to $b$. Thus there is a finite partition of $[a,b]$ on each of whose open components the oscillation of $f$ is below $\eta$. The singleton case is immediate.

On each open component choose one value of $f$, and at every partition point assign the actual value of $f$. The resulting finite step function differs from $f$ by less than $\eta$ everywhere. Taking $\eta =\epsilon \mathrm{/}4$ leaves room for the strengthened construction.

Only finitely many interior breakpoints lie in $E$. Around each such breakpoint $c$, continuity of $f$ gives a small two-sided interval, disjoint from the corresponding intervals for the other breakpoints, on which the oscillation is below $\epsilon \mathrm{/}4$. By [L3] choose a replacement point outside $E$ in that interval and between the neighboring breakpoints. Moving the breakpoint adds only a subinterval from this continuity neighborhood to either adjacent component; the original component oscillation is below $\epsilon \mathrm{/}4$, and a component can be enlarged at both ends, so its new oscillation is at most $\epsilon \mathrm{/}4+\epsilon \mathrm{/}4+\epsilon \mathrm{/}4<\epsilon$, the three pieces overlapping at the old breakpoints. Sampling again and retaining the actual values at all breakpoints and endpoints gives the required approximation with every interior breakpoint outside $E$.