Paper 1, Section II, D

Analysis I | Part IA, 2018

(a) Let q1,q2,q_{1}, q_{2}, \ldots be a fixed enumeration of the rationals in [0,1][0,1]. For positive reals a1,a2,a_{1}, a_{2}, \ldots, define a function ff from [0,1][0,1] to R\mathbb{R} by setting f(qn)=anf\left(q_{n}\right)=a_{n} for each nn and f(x)=0f(x)=0 for xx irrational. Prove that if an0a_{n} \rightarrow 0 then ff is Riemann integrable. If an0a_{n} \rightarrow 0, can ff be Riemann integrable? Justify your answer.

(b) State and prove the Fundamental Theorem of Calculus.

Let ff be a differentiable function from R\mathbb{R} to R\mathbb{R}, and set g(x)=f(x)g(x)=f^{\prime}(x) for 0x10 \leqslant x \leqslant 1. Must gg be Riemann integrable on [0,1][0,1] ?

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