$f+g$ may be a continuous function if

Consider $h\left(x\right)=f\left(x\right)+g\left(x\right)$

If$\text{f(x)}\text{is continuous and}\text{g(x)}\text{is discontinuous}$ , then let us assume that $\text{h(x)}$ is continuous.

Now,       $g\left(x\right)=\underset{\text{continuous}\text{function}}{\underbrace{h\left(x\right)}}-\underset{\text{continuous}\text{function}}{\underbrace{f\left(x\right)}}$

$\Rightarrow g\left(x\right)$ is a continuous function.

Which is contradictory to the given fact that $g\left(x\right)$ is a discontinuous function.

Hence our assumption that $\text{h(x)}$ is continuous is wrong.

i.e., if $f$ is continuous and $g$ is discontinuous then $f+g$ cannot be a continuous function.

i.e., (a) wrong.

Similarly (b) is wrong.

From (c), we take

$f\left(x\right)=x-\left[x\right]\left(\text{Discontinuous}\text{function}\right)$

$g\left(x\right)=x+\left[x\right]\left(\text{Discontinuous}\text{function}\right)$

$(f+g)\left(x\right)=(x-[x\left]\right)+(x+[x\left]\right)$

                $=2x\left(\text{Continuous}\text{function}\right)$