TITLE: Is a continuous function plus a discontinuous function discontinuous?
QUESTION [14 upvotes]: More specifically, I am wondering if $\sin(1/x) - x$ is discontinuous at $0$. I know that $f(x)= \sin(1/x)$ is discontinuous at $0$, but $f(x) = -x$ is continuous at all points. However if I add these two functions together, does that make sum of the function discontinuous?

REPLY [11 votes]: If $f$ is continuous and $h$ is continuous then $g=h -f$ is continuous.
So if $f$ is continuous and $g$ is discontinuous $f+g = h$ can not be continuous else $h-f =g$ would also be continuous.