Paper 1, Section II, E

Analysis I | Part IA, 2017

Let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} be a bounded function defined on the closed, bounded interval [a,b][a, b] of R\mathbb{R}. Suppose that for every ε>0\varepsilon>0 there is a dissection D\mathcal{D} of [a,b][a, b] such that SD(f)sD(f)<εS_{\mathcal{D}}(f)-s_{\mathcal{D}}(f)<\varepsilon, where sD(f)s_{\mathcal{D}}(f) and SD(f)S_{\mathcal{D}}(f) denote the lower and upper Riemann sums of ff for the dissection D\mathcal{D}. Deduce that ff is Riemann integrable. [You may assume without proof that sD(f)SD(f)s_{\mathcal{D}}(f) \leqslant S_{\mathcal{D}^{\prime}}(f) for all dissections D\mathcal{D} and D\mathcal{D}^{\prime} of [a,b].]\left.[a, b] .\right]

Prove that if f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} is continuous, then ff is Riemann integrable.

Let g:(0,1]Rg:(0,1] \rightarrow \mathbb{R} be a bounded continuous function. Show that for any λR\lambda \in \mathbb{R}, the function f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} defined by

f(x)={g(x) if 0<x1λ if x=0f(x)= \begin{cases}g(x) & \text { if } 0<x \leqslant 1 \\ \lambda & \text { if } x=0\end{cases}

is Riemann integrable.

Let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} be a differentiable function with one-sided derivatives at the endpoints. Suppose that the derivative ff^{\prime} is (bounded and) Riemann integrable. Show that

abf(x)dx=f(b)f(a)\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)

[You may use the Mean Value Theorem without proof.]

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