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function and secant intersect

Source: VJIMC 2006 2.3

June 29, 2021
function

Problem Statement

For a function f:[0,1]→Rf:[0,1]\to\mathbb R the secant of ff at points a,b∈[0,1]a,b\in[0,1], a<ba<b, is the line in R2\mathbb R^2 passing through (a,f(a))(a,f(a)) and (b,f(b))(b,f(b)). A function is said to intersect its secant at a,ba,b if there exists a point c∈(a,b)c\in(a,b) such that (c,f(c))(c,f(c)) lies on the secant of ff at a,ba,b.
1. Find the set F\mathcal F of all continuous functions ff such that for any a,b∈[0,1]a,b\in[0,1], a<ba<b, the function ff intersects its secant at a,ba,b. 2. Does there exist a continuous function f∉Ff\notin\mathcal F such that for any rational a,b∈[0,1],a<ba,b\in[0,1],a<b, the function ff intersects its secant at a,ba,b?
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