How to Identify and Prove the Continuity of a Function
We all have heard the word “ continuity” while talking to someone or while reading something. But what if someone asks us the question, what is continuity in maths? The word Continuity comes from “continuous”. It means something that is endless or unbroken or uninterrupted. Therefore we can say that continuity is the presence of a complete path that we can trace on a graph without lifting the pencil. While it is ordinarily true that a continuous function has such graphs, but it won’t be a very precise or practical way to define what is continuity in maths. There are so many graphs and functions that are continuous or connected, in some places, while discontinuous, or broken, in other places. There are functions accommodating too many variables that are to be graphed by hand. Hence, it is extremely necessary that we have a more precise definition of what is continuity in maths. One that does not rely on our expertise to graph and trace a function.
Continuity Of A Function
The continuity of a function at a point can be defined in terms of limits. A function f(x) can be called continuous at x=a if the limit of f(x) as x approaching a is f(a). (A discontinuity can be explained as a point x=a where f is usually specified but is not equal to the limit. Sometimes singularities -- points x=a where f is obscure -- can also be counted as discontinuities.) A continuity of a function on an interval (or some other set) is continuous at each of the single points of that interval (or set). Usually, the term continuity of a function refers to a function that is basically continuous everywhere on its domain.
Conditions for Continuity
In calculus, a continuity of a function can be true at x = a, only if - all three of the conditions below are met:
The function is specified at x = a; i.e. f(a) is equal to a real number
The limit of the function as x addresses a exists
The limit of the function as x addressing a is equal to the function value at x = a
Solved Examples
Question 1) Is the function f(x) continuous at x = 0 in the graph below?
[Image will be uploaded soon]
Solution 1) To check if the function is continuous at x = 0, we also have to check the three conditions:
First, we have to see if the function is defined at x = 0? Yes, f(0) = 2
Second, we have to see if the limit of the function f(x) as x approaches 0 exist? Yes
Lastly, we have to see if the limit of the function f(x) as x approaches 0 equal the function value at x = 0? Yes
Since all of the three conditions have met, we can say that f(x) is continuous at x = 0.
Example 2) Is f(x) continuous at x = -4 in the graph given below?
[Image will be uploaded soon]
Solution 2) To examine for continuity at x = -4, we will have to check the same three conditions:
We have to see if the function is defined; f(-4) = 2
Secondly, we have to check the limit exists
Lastly, the value of the function should not be equal to the limit; point discontinuity at x = 4
FAQs on Continuity in Mathematics: Definitions, Rules & Examples
1. What is the definition of continuity in Maths for Class 12?
In mathematics, continuity refers to a property of a function where a small change in the input results in a correspondingly small change in the output, without any abrupt jumps, breaks, or holes. For a function to be continuous at a point, its graph must be an unbroken curve at that point. Essentially, you can draw the graph of a continuous function without lifting your pen from the paper.
2. What are the three essential conditions for a function f(x) to be continuous at a point x = c?
For a function f(x) to be continuous at a point x = c, it must satisfy all three of the following conditions as per the CBSE syllabus:
The function must be defined at the point c; that is, f(c) must exist and have a finite value.
The limit of the function as x approaches c must exist. This means the Left-Hand Limit (LHL) must be equal to the Right-Hand Limit (RHL).
The value of the function at the point must be equal to the limit of the function at that point. Mathematically, limₓ→_c f(x) = f(c).
3. What are the common types of discontinuity in a function?
Discontinuity occurs when one of the conditions for continuity is not met. The main types are:
Removable Discontinuity: This happens when the limit of the function at a point exists, but is not equal to the function's value at that point, or the function is not defined there. It's like a single “hole” in the graph that could be “plugged”.
Jump Discontinuity: This occurs when the Left-Hand Limit and Right-Hand Limit both exist but are not equal. The function's graph appears to “jump” from one value to another at the point of discontinuity.
Infinite Discontinuity: This occurs when the function's value approaches positive or negative infinity as x approaches a specific point. The graph usually has a vertical asymptote at this point.
4. How can we understand the concept of continuity using a real-world example?
Think about the speed of a car on a journey. If the car accelerates and decelerates smoothly, its speed over time can be represented by a continuous function. The graph of its speed would be an unbroken line. In contrast, consider the cost of mobile data that is sold in 1 GB blocks. The cost remains flat until you use up a block, at which point it suddenly jumps to the next price level. A graph of cost vs. data used would be a series of steps, representing a discontinuous function.
5. What is the difference between a function being 'defined' at a point and being 'continuous' at that point?
A function being 'defined' at a point x = c simply means that f(c) has a value. This is only the first condition for continuity. For a function to be 'continuous' at x = c, it must not only be defined, but its limit must also exist and be equal to f(c). A function can be defined at a point but still be discontinuous if there's a jump or if the limit doesn't match the function's value at that point.
6. Why are polynomial functions considered continuous for all real numbers?
Polynomial functions are always continuous because they are built from the variable x and constants using only addition, subtraction, and multiplication. These operations do not create any divisions by zero, square roots of negative numbers, or other mathematical issues that would cause breaks, gaps, or vertical asymptotes in the graph. As a result, the domain of any polynomial function is all real numbers, and its graph is always a smooth, unbroken curve.
7. What is the importance of the 'Algebra of Continuous Functions'?
The Algebra of Continuous Functions is a set of rules stating that if two functions are continuous, then their sum, difference, product, and quotient (where the denominator is not zero) are also continuous. This is extremely important because it allows us to determine the continuity of complex functions by breaking them down into simpler, known continuous parts (like polynomials, sin(x), cos(x)) without having to check the three conditions of continuity from scratch every time.
8. How do you check for continuity of composite functions?
To check the continuity of a composite function, say g(f(x)), at a point x = c, you must ensure two things. First, the inner function f(x) must be continuous at x = c. Second, the outer function g(x) must be continuous at the point f(c). If both these conditions are met, then the composite function g(f(x)) is guaranteed to be continuous at x = c. This principle is crucial for analysing functions like sin(x²) or log(cos(x)).
