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Romania District MO 2022 Grade 11 P1

Source: Romania District MO 2022 Grade 11

March 27, 2022
darboux s propertyfunctioncollege contestsromania

Problem Statement

Let f,g:R→Rf,g:\mathbb{R}\to\mathbb{R} be functions which satisfy inf⁡x>af(x)=g(a) and sup⁡x<ag(x)=f(a),\inf_{x>a}f(x)=g(a)\text{ and }\sup_{x<a}g(x)=f(a),for all a∈R.a\in\mathbb{R}. Given that ff has Darboux's Property (intermediate value property), show that functions ff and gg are continuous and equal to each other.
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