Continuity and Discontinuity obtained by Algebraic Operations 

1. If $f(x)$ and $g(x)$ are continuous functions in the given interval, the following functions are continuous at $x=a$.
(i) $\mathrm{f}(\mathrm{x}) \pm \mathrm{g}(\mathrm{x})$
(ii) $\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$
(iii) $\frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}$, provided $\mathrm{g}(\mathrm{a}) \neq 0$
2. If $f(x)$ is continuous and $g(x)$ is discontinuous, then $f(x) \pm g(x)$ is a discontinuous function.

Let $f(x)=x$, which is continuous at $x=0$ and $g(x)=[x]$ (greatest integer function) which is discontinuous at $x=0$, are to function.

Now, $\mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x})=\mathrm{x}-[\mathrm{x}]=\{\mathrm{x}\}$ (fractional part of x )
discontinuous at $\mathrm{x}=0$
3. If $f(x)$ is continuous and $g(x)$ is discontinuous at $x=a$ then the product of the functions, $h(x)=f(x)$ $\mathrm{g}(\mathrm{x})$ is may or may not be continuous at $\mathrm{x}=\mathrm{a}$.

 For example,

Consider the functions, $\mathrm{f}(\mathrm{x})=\mathrm{x}^3$. And $\mathrm{g}(\mathrm{x})=\operatorname{sgn}(\mathrm{x})$.
$f(x)$ is continuous at $x=0$ and $g(x)$ is discontinuous at $x=0$
Now,

$
h(x)=f(x) \cdot g(x)=\left\{\begin{array}{cl}
x^3, & x>0 \\
0, & x=0 \\
-x^3, & x<0
\end{array}\right.
$

$h(x)$ is continuous at $x=0$
Take another example, consider $f(x)=x$ and $g(x)=1 /|x|$
$f(x)$ is continuous at $x=0$ and $g(x)$ is discontinuous at $x=0$
Now,

$
h(x)=f(x) \cdot g(x)=x \cdot \frac{1}{|x|}=\operatorname{sgn}(\mathrm{x})
$
And we know that the signum function is discontinuous at $\mathrm{x}=0$.
4. If $f(x)$ and $g(x)$, both are discontinuous at $x=a$ then the the function obtained by algebraic operation of $f(x)$ and $g(x)$ may or may not be continuous at $x=a$.

Consider some examples,

i. $f(x)=[x]$ (Greatest Integer Function) and $g(x)=\{x\}$ (fractional part of $x$ )

Both $f(x)$ and $g(x)$ are discontinuous at $x=1$.
Now, $h(x)=f(x)+g(x)=[x]+\{x\}=x$, which is continuous at $x=1$.
ii. $f(x)=[x]$ (Greatest Integer Function) and $g(x)=\{x\}$ (fractional part of $x$ )

Both $f(x)$ and $g(x)$ are discontinuous at $x=1$.
Now, $h(x)=f(x)-g(x)=[x]-\{x\}=2[x]-x$, which is continuous at $x=1$.
5. Every polynomial function is continuous at every point of the real line.

$
f(x)=a_0 x^n+a_1 x^{n-1}+a_2 x^{n-2}+\ldots . .+a_0 \quad \forall x \in \mathbb{R}
$
6 . Every rational function is continuous at every point where its denominator is not equal to 0
7. Logarithmic, exponential, trigonometric, inverse circular functions; and modulus functions are continuous in their domain.