Answer
False statement
Work Step by Step
We are given:
$f(x)$ discontinuous in $x=c$
$g(x)$ discontinuous in $x=c$
We have to check in $f(x)+g(x)$ is also discontinuous in $c$.
Consider the example:
$f(x)=\begin{cases}
x+1,\text{ for }x\leq 2\\
2x+1,\text{ for }x>2
\end{cases}$
$g(x)=\begin{cases}
-2x+2,\text{ for }x\leq 2\\
-x-2,\text{ for }x>2
\end{cases}$
Compute $f(x)+g(x)$:
$f(x)+g(x)=\begin{cases}
x+1-2x+2,\text{ for }x\leq 2\\
2x+1-x-2,\text{ for }x>2
\end{cases}$
$f(x)+g(x)=\begin{cases}
-x+3,\text{ for }x\leq 2\\
x-1,\text{ for }x>2
\end{cases}$
The functions $f$ and $g$ have a jump discontinuity in $x=2$. We check if $f+g$ is continuous in $x=2$:
$\displaystyle\lim_{x\rightarrow 2^{-}} (f(x)+g(x))=\displaystyle\lim_{x\rightarrow 2^{-}} (-x+3)=1$
$\displaystyle\lim_{x\rightarrow 2^{+}} (f(x)+g(x))=\displaystyle\lim_{x\rightarrow 2^{-}} (x-1)=1$
$(f+g)(2)=-2+3=1$
As the left and right hand limits and the value of $(f_g)$ in $x=2$ are equal, the sum function is continuous in $x=2$.
Therefore the statement is FALSE.
