Discontinuous and Continuous Functions Hi I have a question which I know has already sort of been put onto this site, but I really don't understand how to approach this or whether it is true or false.
I know that if $f$ and $g$ is continuous at $a$, then $f+g$ is continuous at $a$.
Question:
True or False and state why...
If neither $f$ nor $g$ is continuous at $a$, then $f+g$ is not continuous at $a$.
I'm unsure how to substantiate whether the question is true or false and why.
Can anyone help me please?
 A: Consider the following case:
$f(x) =
\begin{cases}
1,  & \text{if $x = 0$} \\[2ex]
0, & \text{if $x \ne 0$}
\end{cases}$
So $f$ is not continuous.
$g(x) =
\begin{cases}
0,  & \text{if $x = 0$} \\[2ex]
1, & \text{if $x \ne 0$}
\end{cases}$
So $g$ is not continuous.
Then $f(x)+g(x) = 1$, which is a continuous function. So the argument is false. And actually this can be generalized:
Let $h(x)$ be a continuous function. Then if $h(0) \ne 0$ we can write $$f(x) =
\begin{cases}
h(x),  & \text{if $x = 0$} \\[2ex]
0, & \text{if $x \ne 0$}
\end{cases}$$ and
$$g(x) =
\begin{cases}
0,  & \text{if $x = 0$} \\[2ex]
h(x), & \text{if $x \ne 0$}
\end{cases}$$
otherwise, if $h(0) = 0$, we can just pick another $x$, e.g. $x = 1$. Then we can write every real valued continuous function $h(x)$ as $h(x) = f(x)+g(x)$ (I am not sure it can be generalized for complex valued functions and maybe some other types of functions that I don't know).
A: Consider a counter example, the discontinuous floor function: $$\left \lfloor{x}\right \rfloor $$
What happens if you do the following?
$$\left \lfloor{x}\right \rfloor - \left \lfloor{x}\right \rfloor $$
