Continuity of the function

Jun 12, 2020, 16:45 IST

Continuity of the function

Definition :-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)
exists (i.e. The right-hand limit = left-hand limit, and both are finite)
exists (i.e. The right-hand limit = left-hand limit, and both are finite)
The function f(x) is said to be continuous in the interval 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 . For a = x1, only the right-hand limit need be considered, and for a = x2, only the left-hand limit needs to be considered.

Example 1 :-Let a function be defined as f(x) = Is this function continuous for all x?

Solution :-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: = 5 – 2 × 1
= 3

-> Right-Hand Limit: = 1 + 2
= 3

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.

Example 2 :-Find b such that f(x) given below is continuous?

Solution :- is a polynomial function and therefore continuous.

is a polynomial function and therefore continuous.
For x = -1

let us consider the left and right hand limits
limit from left of -1

limit from right of -1

for function f to be continuous, we need to have
l1 = l2 = 2 + b
or 2 + b = 1 or b = -1.
Substitute b by -1 in the given function to obtain

the graph of f is shown below and it is clear that the function is continuous at x = -1.

Do solve NCERT text book with the help of Entrancei NCERT solutions for class 12 Maths.