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there exists an interval such that f(I)=[a,b]

Source: romania nmo 2007, grade 11, problem 2

April 15, 2007

functionalgebrapolynomialreal analysisreal analysis unsolved

Problem Statement

Let

f: \mathbb{R}\to\mathbb{R}

be a continuous function, and

a<b

be two points in the image of

f

(that is, there exists

x,y

such that

f(x)=a

and

f(y)=b

). Show that there is an interval

I

such that

f(I)=[a,b]