MathDB
Continuous function

Source: VMO 2019

April 15, 2019
function

Problem Statement

Let f:R→(0;+∞)f:\mathbb{R}\to (0;+\infty ) be a continuous function such that lim⁡x→−∞ f(x)=lim⁡x→+∞ f(x)=0.\underset{x\to -\infty }{\mathop{\lim }}\,f(x)=\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=0. a) Prove that f(x)f(x) has the maximum value on R.\mathbb{R}. b) Prove that there exist two sequeneces (xn),(yn)({{x}_{n}}),({{y}_{n}}) with xn<yn,∀n=1,2,3,...{{x}_{n}}<{{y}_{n}},\forall n=1,2,3,... such that they have the same limit when nn tends to infinity and f(xn)=f(yn)f({{x}_{n}})=f({{y}_{n}}) for all n.n.
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