NCERT Book Class 12 Maths Chapter 5 Continuity and Differentiability

104 MATHEMATICS

Chapter 5
CONTINUITY AND
DIFFERENTIABILITY
v The whole of science is nothing more than a refinement
of everyday thinking.” — ALBERT EINSTEIN v

5.1 Introduction
This chapter is essentially a continuation of our study of
differentiation of functions in Class XI. We had learnt to
differentiate certain functions like polynomial functions and
trigonometric functions. In this chapter, we introduce the
very important concepts of continuity, differentiability and
relations between them. We will also learn differentiation
of inverse trigonometric functions. Further, we introduce a
new class of functions called exponential and logarithmic
functions. These functions lead to powerful techniques of
differentiation. We illustrate certain geometrically obvious
conditions through differential calculus. In the process, we
will learn some fundamental theorems in this area.
Sir Issac Newton
5.2 Continuity (1642-1727)
We start the section with two informal examples to get a feel of continuity. Consider
the function
1, if x ≤ 0
f ( x) = 
2, if x > 0
This function is of course defined at every
point of the real line. Graph of this function is
given in the Fig 5.1. One can deduce from the
graph that the value of the function at nearby
points on x-axis remain close to each other
except at x = 0. At the points near and to the
left of 0, i.e., at points like – 0.1, – 0.01, – 0.001,
the value of the function is 1. At the points near
and to the right of 0, i.e., at points like 0.1, 0.01, Fig 5.1