What's an example of a function ?

Question

$f,g:\mathbb R\to\mathbb R$ isn't a continuous function but $f+g$ is a continuous function .

Give me a hint for an example of a function.

Answer 1

Define

$$f(x) = \begin{cases}1 & x<0\\ -1 & x\geq 0\end{cases}$$

Define $g(x) =-f(x)$ $ \forall x \in \mathbb{R} $

Then $f$ and $g$ are both discontinuous at $0$, but $(f+g)(x) = 0 \forall x \in \mathbb{R}$ and is continuous on $\mathbb{R}$