By continuity of a function, we mean that we can sketch the graph of the function without lifting the pencil. In this article, we will learn the definition of the continuity of a function along with its properties, examples, and solved problems.

Table of Contents

## Definition of Continuity of a Function

Let f(x) be a real-valued function where x is a real number. We say f(x) is continuous at a point x=a if the below holds:

$\lim\limits_{x \to a} f(x)=f(a)$ $\cdots (\star)$

More specifically, if both left-hand and right-hand limit of f(x) exists and is equal to f(a), then we say that f(x) is continuous at x=a, that is,

$\lim\limits_{x \to a-} f(x)=\lim\limits_{x \to a+} f(x)=f(a)$.

For example, the identity function f(x)=x is continuous at x=0 as we have $\lim\limits_{x \to 0} f(x)=f(0)=0$.

Continuity of a function on an interval: A function f(x) is continuous on an interval [a, b] if it is continuous at every point of [a, b]. In this case, the function f(x) is said to be a continuous function on [a, b].

## Definition of Discontinuity

If a function f(x) does not satisfy the above condition $(\star)$, then we say that f(x) is not continuous at x=a, that is, f(x) is a discontinuous function at x=a. In this case, the point x=a is called a discontinuous point of f(x).

For example, f(x)=|x| (modulus of x) is not continuous at x=0. So x=0 is a discontinuous point of the mod function f(x)=|x|.

## Examples of Continuity

Below are a few examples of continuous functions.

- The function f(x)=x
^{n}is continuous for all values of x when n is a positive rational number. If n is a negative number, f(x)=x^{n}is continuous for all x except x=0. - The polynomial function f(x)=a
_{0}x^{n}+a_{1}x^{n-1}+…+a_{n-1}x+a_{n}is continuous for all x. - For all x, the trigonometric functions sinx and cosx are continuous.
- The exponential function f(x)=e
^{x}is continuous for all x. - The logarithmic function f(x)=log
_{e}x is continuous for all x.

## Epsilon Delta Definition of Continuity

Let f(x) be a real-valued function and x=a be a point in the domain of f(x). If for every ε>0 there exists a positive δ (depends on ε) such that

|f(x)-f(a)|<ε whenever 0< |x-a| < δ,

then we say that f(x) is continuous at x=a. This is the epsilon-delta definition of the continuity of a function.

## Properties of Continuity

Below are a few properties that a continuous function satisfies.

- If two functions f(x) and g(x) are continuous at x=a then the following functions are continuous at x=a.
- f(x)+g(x)
- f(x)-g(x)
- f(x) g(x)
- $\dfrac{f(x)}{g(x)}$ when g(a)≠0.

**Intermediate Value Theorem:**If f(x) is a real-valued continuous function on [a, b] and the values f(a) and f(b) are opposite in signs, then there is a point c in [a, b] such that f(c)=0.- Let f(x) is a continuous function on [a, b] and k be a number such that f(a)<k<f(b), then there is a point c in [a, b] such that f(c)=k.
- If f(x) is a continuous function on [a, b], then f(x) must be bounded on [a, b]. The converse is not true, that is, if f(x) is a bounded function, then f(x) may not be continuous.

## Solved Problems on Continuity

**Question 1:** Prove that f(x)=2 is continuous at x=2.

*Solution:*

Note that f(2)=2.

On the other hand, lim_{x→2+}f(x) = 2 and lim_{x→2-}f(x)=2. Thus, we have that

lim_{x→2+}f(x) = lim_{x→2-}f(x) = f(2).

By the definition of continuity, f(x) is continuous at x=2. Note that f(x)=2 is continuous at every point. Actually, constant functions are everywhere continuous.

**Question 2:** Let f(x) be defined by

$f(x)=\begin{cases}1 \quad \quad \text{ if } x>0 \\ 0 \quad \quad \text{ if } x=0 \\ -1 \quad \,\,\text{ if } x<0 \end{cases}$

Discuss the continuity of f(x) at x=0.

*Solution:*

By the definition of f(x), we get that f(0)=0.

lim_{x→0+}f(x) = lim_{x→0+}1 = 1

and

lim_{x→0-}f(x) = lim_{x→0-}-1 = -1.

As lim_{x→0+}f(x) ≠ lim_{x→0-}f(x), we conclude that the function f(x) is not continuous at the point x=0.

## FAQs on Continuity

**Q1: Define continuity of a function with an example.**

Answer: Let f(x) be a function. At x=a, the function f(x) is said to be continuous if the limit of f(x) when x tends to a is equal to f(a). The function f(x)=x^{2} is continuous at x=0.

**Q2: What is discontinuity in Maths?**

Answer: If a function f(x) is not continuous at x=a, then we say that there is a discontinuity of f(x) at x=a. More precisely, if lim_{x→a}f(x) ≠ f(a) then the function f(x) is discontinuous at x=a.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.