Continuity

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What Is Continuity In Maths?

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:

  1. The function is specified at x = a; i.e. f(a) is equal to a real number

  2. The limit of the function as x addresses a exists

  3. 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:

  1. First, we have to see if the function is defined at x = 0? Yes, f(0) = 2

  2. Second, we have to see if the limit of the function f(x) as x approaches 0 exist? Yes

  3. 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:

  1. We have to see if the function is defined; f(-4) = 2

  2. Secondly, we have to check the limit exists

  3. Lastly, the value of the function should not be equal to the limit; point discontinuity at x = 4

FAQ (Frequently Asked Questions)

Question 1) What is Discontinuity?

Answer 1) There are few conditions for a function to be continuous. If these functions are not fulfilled then the function will be discontinuous at that point. Also, there are different types of discontinuity that can be defined on the basis of the failure of the condition.

a) Jump Discontinuity: In jump discontinuity, the right-hand limit as well as the left-hand limit for the function f(x) at x = a exists; but none of them are equal to each other. This can be presented as Lim x→a+f(x)≠Lim x→a−f(x)

b) Infinite Discontinuity: In infinite discontinuity, the function deviates at x = a to become discontinuous in nature. That is to say,

f(a) is not defined

f(a) is not defined

Since the function’s value at x = a tends to be infinite or maybe doesn’t approach an appropriate finite value, the limits of the function as x → a will not be defined.

c) Point Discontinuity: In point discontinuity, the function has a well defined two-sided limit at x = a. The only problem that we might face will be either f(a) will not be defined or f(a) will not be equal to its limit. The discrepancy can be presented as Lim x→af(x)≠f(a)

Question 2) What is a Limit?

Answer 2) The concept of limits plays a commanding role in order to explain the definition of continuity. The limit of a function can be explained as the value of f(x) that the function approaches near a particular value of x.

The limit of the function as x approaches the real number a which from the left can be written like this:

[Image will be uploaded soon]

The limit of a function as x approaches the real number a that from the right can be written like this:

[Image will be uploaded soon]

If the left limit along with the right limit exists (are not infinity) and both of them are equal, then we can say that the limit of the function as x can approach an exists and will be equal to the one-sided limits. We write it like this:

[Image will be uploaded soon]

Remember that the limit describes what a function does very close to a certain value of x. The function value at the point x = a is written f(a).