Let f:[a, b] →ℝ be integrable and ϕ:[m(f), M(f)] →ℝ be continuous. Show that ϕ∘f:[a, b] →ℝ is in

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Let f:[a, b] →ℝ be integrable and ϕ:[m(f), M(f)] →ℝ be...

Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable and $\phi:[m(f), M(f)] \rightarrow \mathbb{R}$ be continuous. Show that $\phi \circ f:[a, b] \rightarrow \mathbb{R}$ is integrable. (Hint: Given $\epsilon>0$, find $\delta>0$ using the uniform continuity of $\phi$. There is a partition $P$ of $[a, b]$ such that $U(P, f)-L(P, f)<\delta^{2}$. Divide the sum in $U(P, f)-L(P, f)$ into two classes depending on whether $M_{i}(f)-m_{i}(f)$ is less than $\delta$, or greater than or equal to $\delta$. Use the Riemann condition for $\phi \circ f$.)

Key Concepts

Partitioning and Oscillation Control

Partitioning the interval into subintervals and analyzing the oscillation of a function within each subinterval is a key technique used to establish integrability. By differentiating between subintervals where the oscillation is small and those where it is larger, one can carefully estimate and control the overall upper and lower sums of the composed function. This approach, aided by the uniform continuity of the outer function, secures the integrability of the composite function.

Function Composition

The composition of a continuous function with an integrable function often retains integrability under appropriate conditions. When a continuous function is uniformly continuous on the range of the integrable function, the small oscillations in the original function are transformed into small oscillations in the composed function. This ensures that the composed function does not introduce wild fluctuations that would disrupt the convergence of Riemann sums.

Uniform Continuity

Uniform continuity on a compact interval means that for every ? > 0 there exists a ? > 0 so that any two points within ? of each other have their function values within ?, independently of where they are located in the domain. This property is especially important because continuous functions on compact sets are automatically uniformly continuous, which allows one to control the variation of the function uniformly over the entire interval.

Riemann Integrability

Riemann integrability is defined by the property that for any given ? > 0, one can find a partition of the interval such that the difference between the upper sum and the lower sum of the function is less than ?. This condition essentially guarantees that the function's behavior is controlled enough over small subintervals so that its overall sum converges to the same limit regardless of how it is approximated by simple functions.

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