A question about monotonic function Suppose that $f,g:\mathbb{R}\to \mathbb{R}$ are both continuous and monotonic , the $f+g$ is monotonic So this clear when both are monotonic and continuous then $f+g$ is monotonic ...--prophetes.ai

A question about monotonic function Suppose that $f,g:\mathbb{R}\to \mathbb{R}$ are both continuous and monotonic , the $f+g$ is monotonic So this clear when both are monotonic and continuous then $f+g$ is monotonic Note: the above is false. but what if neither $f,g$ are monotonic then is $f+g$ is monotonic

It can be. Consider $$ f(x) = \begin{cases} -x, & x\leq 0\\\ 3x,&x>0\end{cases},\qquad g(x) = \begin{cases} 3x, & x\leq 0\\\ -x,&x>0\end{cases} $$ Both $f$ and $g$ are not monotonic, but $(f+g)(x) = 2x$ obviously is.

If $f$ is not monotonic then take $g=f$ and you'll see that $f+g=2f$ is not monotonic, as well.