I'm really stuck on this question. Any help would be appreciated.
2026-04-03 12:50:54.1775220654
Royden Chapter 6
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I think $D_{1/n} f(x)=\frac {f(x+\frac 1n) -f(x)} {1/n}$. For the proof you only have to handle RHS. Write $\int_a^{b} \frac {f(x+\frac 1n) -f(x)} {1/n}$ as $n\int_b^{b+1/n} f(x) dx-n\int_a^{a+1/n} f(x) dx$ and note that this tends to $f(b)-f(a)$ by continuity.