If F, psi:[a,b] rightarrow mathbb{R} with F continuous, psi strictly increasing and (F+psi)(a)>(F+psi)(b) is F+psi decreasing on some interval in (a,b)?

Samson Kaufman 2022-08-12 Answered
Question about strictly increasing and continuous functions on an interval
If F , φ : [ a , b ] R with F continuous, φ strictly increasing and ( F + φ ) ( a ) > ( F + φ ) ( b ) is F + φ decreasing on some interval in (a,b)?
This is clearly true if φ has finitely many points of discontinuity in the interval but I am unsure if this statement is true if there are infinitely many points of discontinuity.
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Answers (1)

kidoceanoe
Answered 2022-08-13 Author has 15 answers
Explanation:
No, not necessarily. The function φ may have a jump discontinuity at each rational number in (a,b), and since F is continuous, F + φ will have exactly the same jump discontinuities, jumping up at each rational. With some details, given any interval (p,q) take a rational r ( p , q ), then lim x r ( F + φ ) ( x ) lim x r + ( F + φ ) ( x ) = lim x r φ ( x ) lim x r + φ ( x ) < 0 so there are s, t near r with s < r < t and ( F + φ ) ( s ) ( F + φ ) ( t ) < 0, that is ( F + φ ) ( s ) < ( F + φ ) ( t ).

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