Continuity definition formula and solved examples in calculus
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?
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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?
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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 of Functions Explained Clearly
1. What is continuity in mathematics?
Continuity means that a function has no breaks, jumps, or holes at a given point or over an interval. In calculus, a function f(x) is continuous at x = a if:
- f(a) is defined.
- lim(x→a) f(x) exists.
- lim(x→a) f(x) = f(a).
2. What is the formal definition of continuity at a point?
A function is continuous at a point if the limit of the function at that point equals the function’s actual value there. Formally, f(x) is continuous at x = a if lim(x→a) f(x) = f(a). This definition is central in calculus and is used to check whether a function has any discontinuity at a specific value.
3. How do you check if a function is continuous at a point?
To check continuity at a point, verify that the limit equals the function value at that point. Follow these steps:
- Step 1: Compute f(a).
- Step 2: Find lim(x→a) f(x).
- Step 3: Compare them.
4. What are the types of discontinuity?
The main types of discontinuity are removable, jump, and infinite discontinuity. These include:
- Removable discontinuity: A hole in the graph where the limit exists but f(a) is not equal to the limit.
- Jump discontinuity: Left-hand and right-hand limits exist but are not equal.
- Infinite discontinuity: The function approaches ±∞ near the point.
5. What is an example of a continuous function?
A polynomial function is an example of a function that is continuous everywhere. For example, f(x) = x² + 3x + 1 is continuous for all real numbers because polynomials have no breaks, holes, or jumps. This means lim(x→a) f(x) = f(a) for every real value of a.
6. Are all polynomial functions continuous?
Yes, all polynomial functions are continuous for every real number. Any function of the form aₙxⁿ + … + a₁x + a₀ has no discontinuities because limits of polynomials always equal their function values. Therefore, polynomials are continuous on (−∞, ∞).
7. Is a rational function always continuous?
A rational function is continuous wherever its denominator is not zero. A rational function has the form f(x) = p(x)/q(x), where q(x) ≠ 0. Discontinuity occurs at values where q(x) = 0, which may create removable or infinite discontinuities.
8. What is continuity over an interval?
A function is continuous over an interval if it is continuous at every point within that interval. This means for every x in the interval, lim(x→a) f(x) = f(a). For example, sin x and eˣ are continuous on (−∞, ∞).
9. What is the difference between continuity and differentiability?
Continuity means no breaks in a function, while differentiability means the function has a defined derivative at a point. Every differentiable function is continuous, but not every continuous function is differentiable. For example, f(x) = |x| is continuous at x = 0 but not differentiable there because the slope changes abruptly.
10. Why is continuity important in calculus?
Continuity is important because many major calculus theorems require functions to be continuous. For example:
- The Intermediate Value Theorem (IVT) applies only to continuous functions.
- The Extreme Value Theorem guarantees maximum and minimum values on closed intervals for continuous functions.
