 # 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 $F,\phi :\left[a,b\right]\to \mathbb{R}$ with F continuous, $\phi$ strictly increasing and $\left(F+\phi \right)\left(a\right)>\left(F+\phi \right)\left(b\right)$ is $F+\phi$ decreasing on some interval in (a,b)?
This is clearly true if $\phi$ 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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Explanation:
No, not necessarily. The function $\phi$ may have a jump discontinuity at each rational number in (a,b), and since F is continuous, $F+\phi$ will have exactly the same jump discontinuities, jumping up at each rational. With some details, given any interval (p,q) take a rational $r\in \left(p,q\right)$, then $\underset{x\to {r}^{-}}{lim}\left(F+\phi \right)\left(x\right)-\underset{x\to {r}^{+}}{lim}\left(F+\phi \right)\left(x\right)=\underset{x\to {r}^{-}}{lim}\phi \left(x\right)-\underset{x\to {r}^{+}}{lim}\phi \left(x\right)<0$ so there are s, t near r with $s and $\left(F+\phi \right)\left(s\right)-\left(F+\phi \right)\left(t\right)<0$, that is $\left(F+\phi \right)\left(s\right)<\left(F+\phi \right)\left(t\right)$.

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