RS Aggarwal Solutions Class 12 Chapter-9 Continuity and Differentiability

Concept of Continuity and Differentiation

Continuity and Differentiation explained in RS Aggarwal Solutions Class 12 Maths Ch 12 is one of the vital concepts for students preparing for board exams. It deals with concepts like the derivative of functions, continuity of certain points, continuity of given intervals, and others. 

The continuity of a function determines the attributes of a function and its functional values.  In a given domain or interval, a function is accepted as continuous if the curve has no breaking points or missing points. That is the curve has to be continuous at every point in its domain. 

A function f(x) is said to be continuous at a point x= y if it meets the following three conditions.

1.  f(y) is continuous if the value of f(y) is finite

2.  f(x) limx→af is continuous at the point if the right-hand limit is the same as that of the left-hand limit. Therefore, R.H.S = L.H.S. 

3.  lim x→af f(x)= f(y)

In a given interval, f(x) can only be continuous if it is equal to x1,x2. All the conditions need to be satisfied with each and every point. 

In differentiation, f(x) is said to be differentiable at the point x = y if the derivative f ‘(y) exists at each point in its interval or domain.

The differentiability formula is given by

f’ (y) = {f(y+h)−f(y)}/ h

State the Differentiability Formula for the Derivatives of the Basic Trigonometric Function

In a particular point, if a function is continuous, then the function can be differentiable at any point x=y, in its domain. However, the vice-versa may not be true every time. 

In the given table, the derivatives of the basic trigonometric functions are explained below from the aspect of differentiability formula: 

Highlights of the RS Aggarwal Solutions for Class 12 Continuity and Differentiability Chapter

• The meaning of continuous functions has been explained in this chapter with reference to graphs. 

• The distinction between continuous and discontinuous functions can be expressed with the help of a graph.

• By practising the solutions given in this chapter, one can understand the proofs of different theorems and the behaviour of continuous functions when these are subjected to algebraic calculations. 

• Students can study several corollaries extracted from theorems and prove them. 

Solved Exercise Questions of RS Aggarwal Class 12 Chapter 9 

Question: 

f(x) = x²

LHL at x = 2

Limx→2 f(x)= Limx→0 f(2-h)

                    = Limx→0 f(2 - h)²

                    = Limx→0 f (h² - 4h + 4)

                    = (0-4) x (0+4)

                    = 4

RHL at x = 2

Limx→2 f(x)= Limx→0 f(2-h)

                    = Limx→0 f (2 - h)²

                    = Limx→0 f (h² - 4h + 4)

                    = (0-4) x (0+4)

                    = 4

Therefore, fx is continuous at 2.

Students studying this maths topic should know about the relationship between differentiability and continuity. This will be an interesting thing to learn in their course of the syllabus.

• If x is a differentiable function, then x has to be continuous. 

• A function can be continuous without it being differentiable.