Continuity and Discontinuity

Introduction

Continuity and discontinuity can be defined as properties of functions that are used in statistics to predict or estimate values. Mathematical functions can be categorised into two types − continuous and discontinuous variables. Continuity is the property of functions that can be shown on a graph without breaking. Discontinuity, on the other hand, is a property of functions which have unconnected points on a graph. In this tutorial, we will understand the concept of continuity and discontinuity along with examples.

Continuity

In nature, continuity is seen all around us. For example, the flow of rivers, time etc., are some of the real-life examples of continuity in nature. In statistics also, we have a continuity of functions having numerical values.

Continuous Functions

Graphically, a continuous function corresponds to a continuous graph. Algebraically, a function f(a) can be called continuous at a point a=x only if it satisfies the following conditions:

• f(x) exists (which means the value of f(x) is finite.

• $\mathrm{\lim_{a \rightarrow x}\:f(a)}$ exists (i.e. both sides of the limit are equal, and both of them are finite)

The function f(a) is said to be continuous if the three conditions mentioned above are satisfied for every point in an interval.

Polynomials

The word polynomials are made up of two words ‘poly’ & ‘nominal’. ‘Poly’ means many & ‘nominal’ means terms. It is an expression consisting of two or more algebraic terms having different power & same variable, these terms are joined together by a mathematical operator.

For example $\mathrm{5x^{2}\:+\:2x\:-\:3}$

Sine and Cosine Function

$$\mathrm{b\:=\:\sin\:a}$$

• The roots of $\mathrm{b\:=\:\sin\:a}$ are found with the multiples of pi(π)

• At $\mathrm{\sin\:a\:=\:0}$, the graph passes through X-axis.

• The period of the sine wave is 2π

$$\mathrm{b\:=\:\cos\:a}$$

• $\mathrm{\sin\:(a\:+\:\pi\:/\:2)\:=\:\cos\:a}$

• $\mathrm{b\:=\:\cos\:a,a\cos}$ function graph is obtained if sin a graph shifted π/2 units to left.

Exponential and Logarithmic Function

An exponential function can be written as $\mathrm{b\:=\:f(a)\:=\:x^{a}}$, where “a” denotes a variable and “x” denotes a constant which is called the base (𝑥 > 1). Number “e” denotes a natural exponential function whose value is 2.71828. The exponential function can be expressed in the form − $\mathrm{y\:=\:e^{x}}$

A logarithm function is the inverse of an exponential function. A logarithm function can be expressed as given below −

Suppose y > 1 is a real number such that the logarithm of a to base y is $\mathrm{ya\:=\:x}$. The logarithm of x to base $\mathrm{y\:\colon\:\log\:_{y}x}$.

Thus $\mathrm{\log\:_{y}x\:=\:a,if\:y^{a}\:=\:x}$

Discontinuity

If a function fails to satisfy the conditions of being a continuous function, then the function is a discontinuous function. The type of discontinuity is determined by the condition that the function fails to satisfy.

Examples of Discontinuous functions and their Point of Discontinuity

The function “f” will be discontinuous at point 𝑎 = 𝑥 in any of the following cases −

• f(x) is not defined.

• $\mathrm{\lim_{a \rightarrow a^{+}}f(a)\:and\:\lim_{a \rightarrow a^{-}}f(a)}$ exist but are not equal.

• $\mathrm{\lim_{a \rightarrow a^{+}}f(a)\:and\:\lim_{a \rightarrow a^{-}}f(a)}$ exist. Both are equal to each other. Both are not equal to f(a).

Greatest and Least Integer functions

Greatest integer function uses the following interpretations −

• [a] = Greatest integer less than equal to “a”

• [a] = Greatest integer not greater than “a”

• [a] = Integral part of “a”

The value of "[a]" can be given by

$$\mathrm{f(a)\:=\:[a]\:=\:n;if\:n\:\leq\:a\leq\:n\:+\:1,n\:\varepsilon\:Z}$$

We interpret the Lowest integer function as −

• [a] = least integer greater than or equal to a

• [a] = least integer, not less than or equal to a

The value of f(a) is an integer (n) such that −

$$\mathrm{f(a)\:=\:n;if\:n\:-\:1\:<a\leq\:n,n\:\varepsilon\:Z}$$

Trigonometric Functions (other than sine and cosine)

Rational Functions

A rational function may be defined as $\mathrm{R(a)\:=\:\frac{P(a)}{Q(a)}}$

Where P(a) and Q(a) are polynomial $\mathrm{Q(a)\neq\:0}$

Signum Function

The Signum function can be defined by the following

$$\mathrm{f(x)\:=\:1,\:if\:x>0}$$

$$\mathrm{f(x)\:=\:1,\:if\:x\:=\:0}$$

$$\mathrm{f(x)\:=\:-1,\:if\:x>0}$$

Where x is a real number, and the range is {−1, 0, 1}

Solved Examples

1)For a function f(a) =

$\mathrm{5\:-\:2a\:\:\:\:\:for\:a<1}$

$\mathrm{3\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:for\:a\:=\:1}$

$\mathrm{a\:+\:2\:\:\:\:\:\:for\:a>1}$

Prove that the function is continuous for all values of a.

Answer − As the function is linear, the function is a straight line on a graph, which means the function is continuous for all a≠ 1. Now for a = 1, we have to check the conditions.

Left-hand limit −

$\mathrm{=\:\lim_{a \rightarrow 1^{-}}\:f(a)}$

$\mathrm{=\:\lim_{a \rightarrow 1^{-}}\:f(5\:-\:2a)}$

$\mathrm{=\:5\:-\:2\times\:1}$

$\mathrm{=\:3}$

Right hand limit −

$\mathrm{=\:\lim_{a \rightarrow 1^{+}}\:f(a)}$

$\mathrm{=\:\lim_{a \rightarrow 1^{+}}\:f(a\:+\:2)}$

$\mathrm{=\:1\:+\:2}$

$\mathrm{=\:3}$

At value a=1

𝑓(1) = 3

As all the conditions are satisfied, we can say that the given function is continuous for all a

2)For a function

$\mathrm{f(a)\:=\:a^{2}\:for\:a<1,}$

$\mathrm{f(a)\:=\:0\:for\:a\:=\:1,}$

$\mathrm{f(a)\:=\:2\:-\:(a\:-\:1)^{2}\:for\:a>1,}$ find the discontinuity.

Answer − The left-hand limit $\mathrm{\lim_{a \rightarrow 1^{-}}f(a)\:=\:1}$ and the right-hand limit $\mathrm{\lim_{a \rightarrow 1^{+}}f(a)\:=\:2}$ We can conclude that the left-hand limit ≠ right-hand limit of the function has a discontinuity at 𝑎 = 1

The point of discontinuity of $\mathrm{f(a)\:=\:a^{2}\:for\:a<1,\:f(a)\:=\:0\:for\:a\:=\:1\:,f(a)\:=\:2\:-\:(a\:-\:1)^{2}\:for\:a>1\:is\:a\:=\:1}$

Conclusion

In this tutorial, we have learned about continuity and discontinuity, along with some functions that are continuous and some that are not, and we learned how to check if a function is continuous or not. A continuous function corresponds to a continuous graph. Continuity is the property of functions that can be shown on a graph without break. Discontinuity, on the other hand, is a property of functions which have unconnected points on a graph. The word polynomials are made up of two words ‘poly’ & ‘nominal’. ‘Poly’ means many & ‘nominal’ means terms.

FAQs

1. What are the real life examples of continuity around us?

Some of the real-life examples of continuity are −

Flow of air in our atmosphere is continuous because it never stops. Growth of hair in our body is a continuous process.

2. Can gender be a continuous variable?

No, gender cannot be considered a continuous variable because it is fixed. It is a discrete variable.

3. Define the range of signum functions?

Range of the signum functions is {−1, 0, 1}

4. What is the standard form of the polynomial with degree 1?

The standard form of a polynomial with degree 1 is $\mathrm{ax\:+\:b}$.

5. What is a zero polynomial?

Zero polynomial is a polynomial whose coefficients are zero.