Finding the supremum of a function given its domain

Question

Let $f:R\to R $ be an increasing function. Take $S$, a non-empty subset of R. Suppose s= sup$S$.Is $f(s)$ the supremum of$~~\left \{f(a)~|~a\in S\right\}$
I tried solving the question this way:
since Sup$S$ = s ;$\forall$a$\in S$, a≤s
since f is increasing f(a)≤f(s) $\implies$f(s) is an upperbound for the given function.
I tried solving further to prove that Supremum is f(s) but things got messy. I finally got the answer after some complicated steps.I need to verify if the answer is correct please help me

Answer 1

Take $$f(x):=\begin{cases} x+1 & x\geq 0\\ x& x<0\end{cases}$$ And $$ S=(-1,0)$$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow$ $\text{sup}(S)=0$

But $$\text{Sup}\{f(a) |~~ a \in S\}=0\neq f(0)$$