I am asking for more explaining about : If we have a function $ F\mathrm{(}x\mathrm{)} $ which is not monotone in $ x $
And then we can say that the function :
$ \overline{F}{\mathrm{(}}{x}{\mathrm{)}}\mathrm{{=}}{F}{\mathrm{(}}{x}{\mathrm{)}}\mathrm{{+}}{Mx} $ For $ {M}\mathrm{{>}}{0} $
is monotone nondecreasing .
Or the function is monotone:
$ \overline{F}{\mathrm{(}}{x}{\mathrm{)}}\mathrm{{=}}{F}{\mathrm{(}}{x}{\mathrm{)}}\mathrm{{-}}{Mx} $ For $ {M}\mathrm{{>}}{0} $
nonincreasing Where the functions are compact and convex functions in closed interval.