TITLE: If $f(x)$ is continuous at $a$ and $g(x)$ is not continuous at $a$, then can $(f+g)(x)$ be continuous at $a$?
QUESTION [6 upvotes]: I know that if both $f(x)$ and $g(x)$ are continuous at $a$, then $(f+g)(x)$ would be continuous at $a$.
My first thought here is that $(f+g)(x)$ cannot be continuous at $a$ if $g(x)$ is not continuous at $a$, but I'm not sure how to prove this.

REPLY [11 votes]: No; if $f+g$ is continuous then $(f+g)-f=g$ must be continuous, which is a contradiction.

REPLY [5 votes]: If $f+g$ is continuous and $f$ is continuous then $(f+g)-f$ is continuous.