Continuity and Discontinuity obtained by Algebraic Operations

Continuity and Discontinuity is one of the important parts of Calculus, which applies to measuring the change in the function at a certain point. Mathematically, it forms a powerful tool by which graphs of functions are determined, the maximum and minimum of functions found, and problems on motion, growth, and decay, to name a few. These concepts of Continuity and Discontinuity have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.

In this article, we will cover the concepts of Continuity and Discontinuity. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023).

Properties of Continuous function

1. If $f, g$ are two continuous functions at a point a of their common domain D. Then $f \pm g$ fg are continuous at a and if $g(a) \neq 0$ then $\underline{f}$
$g$ is also continuous at $\mathrm{x}=\mathrm{a}$.
Suppose $f$ and g be two real functions continuous at a real number c .
Then

(1) $f+g$ is continuous at $x=c$.
(2) $f-g$ is continuous at $x=c$.
(3) $f \cdot g$ is continuous at $x=c$.
$\left(\frac{f}{g}\right)$
(4) $\frac{g}{g}$ is continuous at $\mathrm{x}=\mathrm{c}$, (provided $\mathrm{g}(\mathrm{c}) \neq 0$ ).
The sum, difference, product, and quotient of two continuous functions are always a continuous function. However $h(x)=\frac{f(x)}{g(x)}$ is continous function at $\mathrm{x}=\mathrm{a}$ only if $g(a) \neq 0$

2. If f is continuous at a and $f(a) \neq 0$ then there exists an open interval $(a-\delta, a+\delta$ ) such that for all $x \epsilon(a-\delta, a+\delta) f(x)$ has the same sign as $f(a)$

3. If a function $f$ is continuous on a closed interval $[a, b]$, then it is bounded on ( $\mathrm{a}, \mathrm{b}$ ) and there exists real numbers k and K such that $k \leq f(x) \leq K$ for all $x \in[a, b]$

Continuity and Discontinuity obtained by Algebraic Operations

1. If $f(x)$ and $g(x)$ are continuous functions in the given interval, then the following functions are continuous at $\mathrm{x}=\mathrm{a}$.
   (i) $f(x) \pm g(x)$
   (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, $f(x)-g(x)=x-[x]=\{x\}$ (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, $h(x)=f(x) g(x)$ is may or may not be continuous at $x=a$.

   For example,
   Consider the functions, $f(x)=x^3$. And $g(x)=\operatorname{sgn}(x)$.
   $f(x)$ 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 signum function is discontinuous at $x=0$.

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,

1. $f(x)=[x]$ (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 $x=1$.

2. $f(x)=[x]$ (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 $x$ $=1$.

3. 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 \ldots+a_0 \forall x \in \mathbb{R}
   $

4. Every rational function is continuous at every point where its denominator is not equal to $0$

5. Logarithmic functions, exponential functions, trigonometric functions, inverse circular functions; and modulus functions are continuous in their domain.

Solved Examples Based On the Continuity and Discontinuity obtained by Algebraic Operations :

Example 1: Let $f(x)=\sin x \quad \forall x \epsilon e$ and $g(x)=\cos x \quad \forall x \in R$ then which of the following is not continuous?
1) $f(x)+g(x)$
2) $f(x)-g(x)$
3) $f(x)^{\star} g(x)$
4) $f(x) / g(x)$

Solution:
Properties of continuous function -
If $f, g$ are two continuous functions at a point a of their common domain D . Then $f \pm g$ fg are continuous at a and if $g(a) \neq 0$ then
$\underline{f}$
$g \quad$ is also continuous at $\mathrm{x}=\mathrm{a}$.

$\because f(x)=\sin x, g(x)=\cos x$ are continuous for all $x \epsilon R$
$\therefore f(x) \pm g(x)$ and $\therefore f(x) \cdot g(x)$ will also be continuous for all $x \in R$
$\frac{f(x)}{g(x)}$ will be discontinuous whenever $\mathrm{g}(\mathrm{x})=0$, so there are various x in $(-\infty, \infty)$ such that $g(x)=\cos x=0$
$\therefore \frac{f(x)}{g(x)}$ is not continuous throughout R .
Hence, the answer is the option 4.

Example 2: Let $f(x)=x+1 \forall x \in R$ and $g(x)=x^2-3 x+2$ $\frac{f(x)}{g(x)}$ equals
1) $2$
2) $1$
3) $0$
4) $3$

Solution:

Properties of Continuous function -

If $f, g$ are two continuous functions at a point a of their common domain D . Then $f \pm g$ fg are continuous at a and if $g(a) \neq 0$ then $\underline{f}$ $g$ is also continuous at $\mathrm{x}=\mathrm{a}$.

$
\frac{f(x)}{g(x)}=\frac{x+1}{x^2-3 x+2}
$

which is not defined when $x^2-3 x+2=0$
$\Rightarrow x=1,2$ so discontinuous at 2 points.
Hence, the answer is the option 1

Example 3: Which of the following is true?
1) If $f(x)$ is continuous at $1, f(1)=10$ then there exists an interval $(1-\delta, 1+\delta)$ such that $\forall x \in(1-\delta, 1+\delta), f(x)=-5$
2) If $f(x)$ is continuous at $1, f(1)=10$ then there exists an interval $(1-\delta, 1+\delta)$ such that $\forall x \in(1-\delta, 1+\delta), f(x)=1$
3) If $f(x)$ is continuous at $1, f(1)=10$ then there exists an interval $(1-\delta, 1+\delta)$ such that $\forall x \epsilon(1-\delta, 1+\delta), f(x)>0$
4) If $f(x)$ is continuous at $1, f(1)=10$ then there exists an interval $(1-\delta, 1+\delta)$ such that $\forall x \epsilon(1-\delta, 1+\delta), f(x)<0$

Solution:

Properties of continuous function -

If f is continuous at a and $f(a) \neq 0$ then there exists an open interval ( $a-\delta, a+\delta$ ), such that all have the same sign as $f(a)$ If $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=\mathrm{a}, \mathrm{f}(\mathrm{a}) \neq$ zero then there exists an interval $(a-\delta, a+\delta)$ such that $\forall n \epsilon(a-\delta, a+\delta), \mathrm{f}(\mathrm{x})$ has the same sign as $f(a)$.

Hence, the answer is the option (3).

Example 4: Let $f:[-1,3] \rightarrow R_{\text {be defined as }}$

$
f(x)=\left\{\begin{array}{rlrl}
|x|+[x], & & -1 & \leq x<1 \\
x+|x|, & & 1 & \leq x<2 \\
x+[x], & & 2 \leq x \leq 3\\
\end{array}\right.
$

${\text {where }[t] \text { denotes the greatest }}$integer less than or equal to $t$. Then $f$ is discontinuous at:

1) only one point
2) only two points
3) only three points
4) four or more points

Solution:

Continuity of composite functions-
A composite function $f \circ g(x)$ is continuous at $x=a$ if $g$ is continuous at $x=$ a and f is continuous at $\mathrm{g}(\mathrm{a})$.

$
\begin{aligned}
& f(x)=\left\{\begin{array}{rc}
|x|+[x], & -1 \leq x<1 \\
x+|x|, & 1 \leq x<2 \\
x+[x], & 2 \leq x \leq 3
\end{array}\right. \\
& f(x)=\left\{\begin{array}{cc}
-x-1, & -1 \leq x<0 \\
x+0, & 0 \leq x<1 \\
2 x, & 1 \leq x<2 \\
x+2, & 2 \leq x<3 \\
x+3, & x=3
\end{array}\right.
\end{aligned}
$

$f(x)$ is discontinuous at $x=0,1,3$.
Hence, the answer is the option (3).

Example 5 : Which of the following functions is not bounded in $(0,5)$ ?
1) $\sin x$
2) $\cos x$
3) $1 / x$
4) $x^2$

Solution:
Properties of Continuous function -
If a function $f$ is continuous on a closed interval [ $a, b]$ then it is bounded on ( $a$, b) there exist real numbers $k$ and $K$ such that

$
k \leq f(x) \leq K \text { for all } x \in[a, b]
$
In $[0,5] \sin x, \cos x, x^2$ All are continuous, only $1 / \mathrm{x}$ is not continuous.
Hence, the answer is the option 3.

Summary

Continuity and Discontinuity is an important concept in Calculus. It provides a deep understanding of how the functions interact and change. It is very helpful in practical applications for physics, economics, etc. There are various kinds of discontinuity at a point. Overall, this provides a better interpretation of the functions leading to more accurate solutions.