Paper 1, Section II, E

Analysis I | Part IA, 2014

(i) State the Mean Value Theorem. Use it to show that if f:(a,b)Rf:(a, b) \rightarrow \mathbb{R} is a differentiable function whose derivative is identically zero, then ff is constant.

(ii) Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} be a function and α>0\alpha>0 a real number such that for all x,yRx, y \in \mathbb{R},

f(x)f(y)xyα.|f(x)-f(y)| \leqslant|x-y|^{\alpha} .

Show that ff is continuous. Show moreover that if α>1\alpha>1 then ff is constant.

(iii) Let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} be continuous, and differentiable on (a,b)(a, b). Assume also that the right derivative of ff at aa exists; that is, the limit

limxa+f(x)f(a)xa\lim _{x \rightarrow a+} \frac{f(x)-f(a)}{x-a}

exists. Show that for any ϵ>0\epsilon>0 there exists x(a,b)x \in(a, b) satisfying

f(x)f(a)xaf(x)<ϵ.\left|\frac{f(x)-f(a)}{x-a}-f^{\prime}(x)\right|<\epsilon .

[You should not assume that ff^{\prime} is continuous.]

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