Continuity of Functions: Definition, Solved Examples

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.

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)=xⁿ is continuous for all values of x when n is a positive rational number. If n is a negative number, f(x)=xⁿ is continuous for all x except x=0.
• The polynomial function f(x)=a₀xⁿ+a₁x^n-1+…+a_n-1x+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_ex 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.

1. 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.
2. 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.
3. 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.
4. 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