Question about strictly increasing and continuous functions on an interval

If $F,\phi :[a,b]\to \mathbb{R}$ with F continuous, $\phi $ strictly increasing and $(F+\phi )(a)>(F+\phi )(b)$ 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.

If $F,\phi :[a,b]\to \mathbb{R}$ with F continuous, $\phi $ strictly increasing and $(F+\phi )(a)>(F+\phi )(b)$ 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.