MathLabs

Problem 4

A point in the plane with a Cartesian coordinate system is called a mixed point if one coordinate is rational and the other is irrational. Find all polynomials with real coefficients whose graphs contain no mixed point.
Step 5 of 7: Apply the intermediate value theorem
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f(1)−q<0<f(p)−q  ⟹  f(r)=q for some r∈(1,p).f(1)-q<0<f(p)-q\implies f(r)=q\text{ for some }r\in(1,p).
Detailed analysis

Continuity gives a real root r of f(x)-q between 1 and p.