Paper 1, Section II, F

Analysis I | Part IA, 2010

(a) Let n1n \geqslant 1 and ff be a function RR\mathbb{R} \rightarrow \mathbb{R}. Define carefully what it means for ff to be nn times differentiable at a point x0Rx_{0} \in \mathbb{R}.

 Set sign(x)={x/x,x00,x=0.\text { Set } \operatorname{sign}(x)= \begin{cases}x /|x|, & x \neq 0 \\ 0, & x=0 .\end{cases}

Consider the function f(x)f(x) on the real line, with f(0)=0f(0)=0 and

f(x)=x2sign(x)cosπx,x0.f(x)=x^{2} \operatorname{sign}(x)\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 .

(b) Is f(x)f(x) differentiable at x=0x=0 ?

(c) Show that f(x)f(x) has points of non-differentiability in any neighbourhood of x=0x=0.

(d) Prove that, in any finite interval II, the derivative f(x)f^{\prime}(x), at the points xIx \in I where it exists, is bounded: f(x)C\left|f^{\prime}(x)\right| \leqslant C where CC depends on II.

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