Is it possible that if $f(x)$ is continuous at $0$ and $g(x)$ is NOT continuous at $0$ then $f(x) + g(x)$ is continuous at $0$?

Question

I have to determine whether it is true that if $f(x)$ is continuous at $0$ and $g(x)$ is NOT continuous at $0$ then $f(x) + g(x)$ is continuous at $0$ (both functions live in $R^{1}$. Call this statement 1. I want to use the contrapositive of the Algebraic Continuity Theorem to show that statement 1 is false. Here is my argument so far:

Note the contrapositive of Statement 1:

$f(x) + g(x)$ is NOT continuous at 0 $\rightarrow$ $f(x)$ is NOT continuous at $0$ OR $g(x)$ is continuous at $0$.

Now, note the contrapositive of (the addition version of) Algebraic Continuity Theorem: $f(x) + g(x)$ is NOT continuous at 0 $\rightarrow$ $f(x)$ is NOT continuous at $0$ OR $g(x)$ is NOT continuous at $0$ which contradicts statement 1.

I don't think this argument correct. Is it possible to modify this argument to make it prove what it is suppose to prove?

Answer 1

The differnce $g(x)=(f(x)+g(x))-f(x)$ of two continuous functions is continuous