Introduction to the Continuity of a function 

In mathematics, the continuity of a function is a property which describes the behaviour of the function at every point in its domain. It is an important property for a function to be differentiable. Let us learn more about continuity of a function and its definition.

Understanding of the Continuity of a Function

The term continuity refers to a consistent or unbroken operation that does not experience any disturbance. In mathematics, continuity is a property of a function that decides its behaviour on a specific domain. A function is known as a continuous function if it is defined on every point in its domain. The continuity of a function is defined as:

“A function f(x) is said to be a continuous function at a point c if there is no disturbance in the graph of f(x) then the limit of the function at c must exist and the value of the limit and the function at c should be equal.”

For example, the flow of water in a straight tunnel is continuous. But if the tunnel is not straight then the flow of water will be discontinuous. Similarly, a function that does not have the ability to be defined on every point in its domain, is known as a discontinuous function.

Continuity of a Function Definition and Formula 

A function f(x) is said to be a continuous function over an open interval (a, b) if it satisfies the following conditions.

• If there exists a point c in the open interval (a, b).
• $\lim_{x\to c} f(x) = f(c)$
• $f(a) = f(b) = f(c)$

Or, a function is continuous on (a, b) if the limit of the function is equal to the value of the function at that point. Let’s understand how we can determine if a function is continuous or discontinuous. 

How do you find the continuity of a function?

You can determine either the function is continuous or discontinuous by following some simple steps. These steps are:

1. Identify the function for which you want to determine continuity. 
2. Find the value of the function at the given interval i.e., on (a, b).
3. Now calculate the limit of the function at both points a and b. 
4. If the limit from x to a of f(x) is equal to the f(a) then the function is continuous. 
5. If the limit from x to b of f(x) is equal to the f(b) then the function is continuous. 
6. The conditions in step 5 and 6 should be satisfied. Otherwise, the function will not be considered as a continuous function. 

Continuity of a function example 1

Let’s discuss the continuity of the function $f(x)=\frac{x^2-x-2}{x+1}$ at $x = 2$.

To discuss the continuity of the given function, we have to find the value of the function at the given point. 

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

At x = 2

$f(2)=\frac{2^2 – 2 – 2}{2+1} = \frac{4 – 4}{3}$

$f(2) = \frac{0}{3} = 0$

Now evaluating the right and left hand side limit of the function at +2 and -2. 

$\lim{x\to 2+} f(x)=\lim{x\to 2+}\frac{x^2 – x – 2}{x + 1}$

$\lim{x\to 2+} f(x)=\frac{2^2 – 2 – 2}{2+1} = \frac{4 – 4}{3} = 0$

Similarly, 

$\lim{x\to 2-} f(x)=\lim{x\to 2-} \frac{x^2 – x – 2}{x + 1}$

$\lim{x\to 2-} f(x)= \frac{2^2 – 2 – 2}{2+1} = \frac{4 – 4}{3} = 0$

Since the value of the function and limit at point x=2 is equal. Therefore, the function is continuous at this point.

Continuity and Differentiability of a function

In calculus, the differentiability of a function depends on the continuity. It is stated as the rate of change in the function due to the change in its variable. For a continuous function over an open interval (a, b), there exists a point c in (a, b) then, if $\lim_{h\to 0}\frac{f(c+h) –f(c)}{h}$ exists for every c in (a, b), the function f(x) is known as differentiable function. Mathematically,

$f’(c)=\lim_{h\to 0}\frac{f(c+h) – f(c)}{h}$

Where, f’(c) is known as the derivative of f(x) at point c. If f’(c) is zero, the function will satisfy Rolle's Theorem.

In other words, if the slope of the tangent line is equal to the limit of the function at that point, it is called differentiability of a function. A function should be continuous to be differentiable. But sometimes, a discontinuous function is also differentiable at some points in its domain. 

Conclusion

In calculus, the continuity of a function is the smoothness and consistent behaviour of the function. A function is continuous if there exists a point in the open interval such that the limit of the function and its value on this point is equal. Moreover, a function must be continuous to be differentiable. Otherwise it will be known as a discontinuous function.