Question

Let $f\left(x\right)$ be a continuous and $g\left(x\right)$
be a discontinuous function. Prove that $f\left(x\right)+g\left(x\right)$ is a discontinuous
function.

Answer 1

Let g(x)be continuous at x =a $\Rightarrow \underset{x\to a}{lim}g\left(x\right)\ne g\left(a\right)$

Now, $\underset{x\to a}{lim}\left[f\right(x)+g(x\left)\right]=\underset{x\to a}{lim}f\left(x\right)+\underset{x\to a}{lim}g\left(x\right)$ would not

be equal to $f\left(a\right)+g\left(a\right)$ has $\underset{x\to a}{lim}g\left(x\right)\ne g\left(a\right)\Rightarrow$ Discontinuous at x = a.