Paper 1, Section II, 12F

Analysis I | Part IA, 2016

Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} satisfy f(x)f(y)xy|f(x)-f(y)| \leqslant|x-y| for all x,y[0,1]x, y \in[0,1].

Show that ff is continuous and that for all ε>0\varepsilon>0, there exists a piecewise constant function gg such that

supx[0,1]f(x)g(x)ε.\sup _{x \in[0,1]}|f(x)-g(x)| \leqslant \varepsilon .

For all integers n1n \geqslant 1, let un=01f(t)cos(nt)dtu_{n}=\int_{0}^{1} f(t) \cos (n t) d t. Show that the sequence (un)\left(u_{n}\right) converges to 0 .

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