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Continuity and Differentiability

Continuity

The property of continuity is exhibited by various aspects of nature. The water flow in the rivers is continuous. The flow of time in human life is continuous i.e. you are getting older continuously. And so on. Similarly, in mathematics, we have the notion of the continuity of a function.

What it simply means is that a function is said to be continuous if you can sketch its curve on a graph without lifting your pen even once (provided that you can draw good). It is a very straightforward and close to accurate definition actually. But for the sake of higher mathematics, we must define it in a more precise way. That’s what we are going to do in this section. So let’s jump into it!

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Definition of Continuity

A function f(x) is said to be continuous at a point x = a, in its domain if the following three conditions are satisfied:

  1. f(a) exists (i.e. the value of f(a) is finite)
  2. Limx→a f(x) exists (i.e. the right-hand limit = left-hand limit, and both are finite)
  3. Limx→a f(x) = f(a)

The function f(x) is said to be continuous in the interval I = [x1,x2] if the three conditions mentioned above are satisfied for every point in the interval I.

However, note that at the end-points of the interval I, we need not consider both the right-hand and the left-hand limits for the calculation of Limx→a f(x). For a = x1, only the right-hand limit need be considered, and for a = x2, only the left-hand limit needs to be considered.

Browse more Topics under Continuity And Differentiability

Some Typical Continuous Functions

  • Trigonometric Functions in certain periodic intervals (sin x, cos x, tan x etc.)
  • Polynomial Functions (x2 +x +1, x4 + 2….etc.)
  • Exponential Functions (e2x, 5ex etc.)
  • Logarithmic Functions in their domain (log10x, ln x2 etc.)

Discontinuity

If any one of the three conditions for a function to be continuous fails; then the function is said to be discontinuous at that point. On the basis of the failure of which specific condition leads to discontinuity, we can define different types of discontinuities.

Jump Discontinuity

In this type of discontinuity, the right-hand limit and the left-hand limit for the function at x = a exists; but the two are not equal to each other. It can be shown as: $$ { Lim_{x\rightarrow{a^+}}  f(x) \neq  Lim_{x\rightarrow{a^-}}  f(x) } $$

continuity

Infinite Discontinuity

The function diverges at x = a to give it a discontinuous nature here. That is to say, $$ {\text{f(a) is not defined}} $$ Since the value of the function at x = a tends to infinity or doesn’t approach a particular finite value, the limits of the function as x → a are also not defined.

continuity

Point Discontinuity

This is a category of discontinuity in which the function has a well defined two-sided limit at x = a, but either f(a) is not defined or f(a) is not equal to its limit. The discrepancy can be shown as: $$ { Lim_{x\rightarrow{a}}  f(x) \neq f(a) } $$ This type of discontinuity is also known as a Removable Discontinuity since it can be easily eliminated by redefining the function in such a way that, $$ { f(a) = Lim_{x\rightarrow{a}}  f(x) } $$

continuity

Solved Examples for You

Question 1: Let a function be defined as f(x) =

5 – 2x for x < 1
3 for x = 1
x + 2 for x > 1

Is this function continuous for all x?

Answer : Since for x < 1 and x > 1, the function f(x) is defined by straight lines (that can be drawn continuously on a graph), the function will be continuous for all x ≠ 1. Now for x = 1, let us check all the three conditions:

–> f(1) = 3 (given)

–> Left-Hand Limit:
\( {= Lim_{x\rightarrow{1^-}}  f(x)} \)
\( {= Lim_{x\rightarrow{1^-}} (5 – 2x)} \)
\( {= 5 – 2\times{1}} \)
\( {= 3} \)

–> Right-Hand Limit:
\( {= Lim_{x\rightarrow{1^+}}  f(x)} \)
\( {= Lim_{x\rightarrow{1^+}} (x + 2)} \)
\( {= 1 + 2} \)
\( {= 3} \)

–> \( {Lim_{x\rightarrow{1^-}}  f(x) = Lim_{x\rightarrow{1^+}}  f(x) = 3 = f(1)} \)

Thus all the three conditions are satisfied and the function f(x) is found out to be continuous at x = 1. Therefore, f(x) is continuous for all x.

This concludes our discussion on the topic of continuity of functions. Continuous functions are very important as they are necessarily differentiable at every point on which they are continuous, and hence very simple to work upon.

Question 2: Explain the three conditions of continuity?

Answer: The three conditions of continuity are as follows:

  • The function is expressed at x = a.
  • The limit of the function as the approaching of x takes place, a exists.
  • The limit of the function as the approaching of x takes place, a is equal to the function value f(a).

Question 3: What is limit with regards to continuity?

Answer: A limit refers to a number that a function approaches as the approaching of an independent variable of the function takes place to a given value. For example, given the function f (x) = 3x, the limit of f (x) as the approaching of x takes place to 2 is 6. Symbolically, one can write this as f (x) = 6.

Question 4: What is meant by discontinuous function?

Answer: Discontinuous functions are those that are not a continuous curve. In a removable discontinuity, one can redefine the point so as to make the function continuous by matching the particular point’s value with the rest of the function.

Question 5: Is a point function continuous?

Answer: A point function is not continuous according to the definition of continuous function.

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