Question about strictly increasing and continuous functions on an intervalIf F , φ : [ a , b ] → R with F continuous, φ strictly increasing and ( F + φ ) ( a ) &gt; ( 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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Question about strictly increasing and continuous functions on an intervalIf   F  ,  φ  :  [  a  ,  b  ]  →      R   with F continuous,   φ strictly increasing and   (  F  +  φ  )  (  a  )  &amp;gt;  (  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.

kidoceanoe · Accepted Answer

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      )      &amp;lt;      0       so there are s, t near r with   s  &amp;lt;  r  &amp;lt;  t and   (  F  +  φ  )  (  s  )  −  (  F  +  φ  )  (  t  )  &amp;lt;  0, that is   (  F  +  φ  )  (  s  )  &amp;lt;  (  F  +  φ  )  (  t  ).

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)?

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