Question

Let $f\left(x\right)=\left[x\right]$ and $g\left(x\right)=sgn\left(x\right)$ (where $[\cdot ]$ denotes greatest integer function), then discuss the continuity of $f\left(x\right)\pm g\left(x\right)$, $f\left(x\right).g\left(x\right)$ and $\frac{f\left(x\right)}{g\left(x\right)}$ at $x=0$.

Answer 1

$f\left(x\right)=\left[x\right]$
We know that greatest integer function is discontinuous at all integers.
So, f(x) is discontinuous at $x=0$
$g\left(x\right)=sgn\left(x\right)$
Signum function is not defined at $x=0$
So the function is discontinuous at $x=0$
We know that sum,difference ,product and quotient (defined in its domain)of two continuous functions is continuous.
So, $f\pm g,fg,\frac{f}{g}$ are all discontinuous at $x=0$