# JEE Main 2020 Study Notes Continuity and Differentiability

By Subrato Banerjee|Updated : July 5th, 2019

Continuity and Differentiability are the properties of functions and applications of limit. The questions set up on this topic are extensively asked in JEE Main / JEE Advanced /BITSAT. One can expect 3-5 questions directly from these topics and another 2-3 question in extension with other topics such as in algebra and coordinate geometry. Every candidate should master these topics to gain easy and definite marks.

## 1. CONTINUITY

A function f(x) is said to be continuous if the function

(a). Satisfies the definition of limit

(b). And Left-hand limit = Right-hand limit = f(value at that limit)

### Explanation of definition

Whenever a function satisfies the above condition, then it is imagined that at the desired point if there would have been a total complete curve without having any point of discontinuity

The general expression  As shown above, the point of the break is the point where there is a suspect that there might be a discontinuity or break in continuity.

So, we need to check at this point

Illustration 5:

Check continuity of the function at x=3 and x=-3 Ans.

Now checking for x=3  cancelling (x-3) from numerator and denominator

Thus, we get Thus, putting x= 3, we get

= 27 / 6 = 9 / 2

Also,  cancelling (x-3) from numerator and denominator

Thus, we get Thus, putting x=3, we get

= 27/6 or 9/2

f(3)=27/6 or 9/2

Hence it satisfies the definition of continuity at x=3 hence continuous at x=3

For x=-3 =  cancelling (x-3) from numerator and denominator

Thus, we get Thus, putting x=-3, we get cancelling (x-3) from numerator and denominator

Thus, we get Thus, putting x=-3, we get f (3) =∞

Though all three gives undefined value since they are not finite, we cannot compare. Hence they do not satisfy the definition of continuity.

Thus, x=-3 is discontinuous.

## 1.1 Graphical demonstration of a continuous function graphically The given graph depicts f(x) =x3 - 0.25

And there is a break at x=2

Thus, if the curve needs to be continuous it must have a definite value at 2 such that it will follow the properties of the curve.

## 2. DIFFERENTIABILITY

According to the general definition of differentiability, it means the change in a certain property which is dependent on certain other property to the change of another property.

Mathematically it can be represented as follows

### 2.1 Ways of representing differentiability

The differentiation of a function of x with respect to x is represented

(a). f' (x)

(b). D(f(x)): where D is d / dx

(c). d(f(x) / dx

### 2.2 Conditions for a function to be differentiable

(i) The relation given must be a function

(ii) The function must have a certain limiting condition

(iii) The limit must exist at the limiting point

(iv) The limit must be continuous or else must contain a removable discontinuity

(v) If the limit contains removable discontinuity, then it can only be partially differentiable

(vi) If only the limit is continuous, then it can be differentiable

NOTE: If a limit is differentiable then surely it is continuous, but the converse is not necessarily true.

Illustration 6:

Let Check whether differentiable or not

Ans.

Checking for continuity Thus Hence the function is continuous)

Now checking whether differentiable or not

LHD: RHD: As, LHD = RHD

Hence, the given function is differentiable at x=1

### 2.3 Differentiable over an interval

(i) F(x) is said to be differentiable over an open interval (a, b) if it is differentiable at each point of the open interval (a, b)

(ii) F(x) is said to be differentiable over [a, b] if:

(iii) F(x) is differentiable in (a, b) and

(iv) For the points a and b, f(a+) and f(b-) exist

More from us:

JEE Main Syllabus with weightage

JEE Main Question Paper 2019 with Solutions

Prepare Smartly. The Most Comprehensive Exam Prep App GradeStack Learning Pvt. Ltd.Windsor IT Park, Tower - A, 2nd Floor, Sector 125, Noida, Uttar Pradesh 201303 help@byjusexamprep.com