Supremum of the image of a monotonic function

Question

For a monotonic function $f:D\to\mathbb{R}$ where $D\subset\mathbb{R}$, and two sets $A$ and $B$ such that $\sup A=\sup B$, is it true that $\sup f(A)=\sup f(B)$, where $f(A)$ denotes the image of set $A$ under function $f$? If so, how to prove it, assuming $f(\sup A)$ and $f(\sup B)$ may be undefined?

Answer 1

No. This need not be true. The function may be monotonic but it may be decreasing. e.g. y= -x and choose D= [0,1] A =[1/2, 1] B=[1/3, 1] then Sup(A)=Sup(B)=1 and next thing is reversed. Right?