Paper 1, Section II, F

Analysis I | Part IA, 2014

Define what it means for a function f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} to be (Riemann) integrable. Prove that ff is integrable whenever it is

(a) continuous,

(b) monotonic.

Let {qk:kN}\left\{q_{k}: k \in \mathbb{N}\right\} be an enumeration of all rational numbers in [0,1)[0,1). Define a function f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} by f(0)=0f(0)=0,

f(x)=kQ(x)2k,x(0,1]f(x)=\sum_{k \in Q(x)} 2^{-k}, \quad x \in(0,1]


Q(x)={kN:qk[0,x)}Q(x)=\left\{k \in \mathbb{N}: q_{k} \in[0, x)\right\}

Show that ff has a point of discontinuity in every interval I[0,1]I \subset[0,1].

Is ff integrable? [Justify your answer.]

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