Paper 1, Section II, F

Analysis I | Part IA, 2018

(a) Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} be a function, and let xRx \in \mathbb{R}. Define what it means for ff to be continuous at xx. Show that ff is continuous at xx if and only if f(xn)f(x)f\left(x_{n}\right) \rightarrow f(x) for every sequence (xn)\left(x_{n}\right) with xnxx_{n} \rightarrow x.

(b) Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} be a non-constant polynomial. Show that its image {f(x):xR}\{f(x): x \in \mathbb{R}\} is either the real line R\mathbb{R}, the interval [a,)[a, \infty) for some aRa \in \mathbb{R}, or the interval (,a](-\infty, a] for some aRa \in \mathbb{R}.

(c) Let α>1\alpha>1, let f:(0,)Rf:(0, \infty) \rightarrow \mathbb{R} be continuous, and assume that f(x)=f(xα)f(x)=f\left(x^{\alpha}\right) holds for all x>0x>0. Show that ff must be constant.

Is this also true when the condition that ff be continuous is dropped?

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