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 Incoming links: Continuous approximation of a Riemann-integrable function

Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 12E / c / Solution Created 2026-09-24 Updated 2026-10-03

The continuous approximation of a Riemann-integrable function follows here by choosing partitions

P_{n}

with upper-minus-lower sum below

1/ n

. On each subinterval choose the midpoint of the infimum and supremum to form a step function

s_{n}

. Then

\int_{a}^{b} ∣ g - s_{n} ∣ < \frac{1}{n} .

Replace each of the finitely many jumps of

s_{n}

by a linear transition on intervals of sufficiently small total length, obtaining a continuous

ϕ_{n}

with

\int ∣ s_{n} - ϕ_{n} ∣ < 1/ n

. Hence

\int ∣ g - ϕ_{n} ∣ < 2/ n

, and uniformly for every subinterval

[α, β]

,

\int_{α}^{β} (g - ϕ_{n}) \leq \int_{a}^{b} ∣ g - ϕ_{n} ∣ \to 0.

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