If for find the value which can be assigned to f(x) at x = π/4 so that the function f(x) becomes continuous every where in [0, π/2].

Basic Concept:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if :

where h is a very small ‘+ve’ no.

i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if :

Function is defined for [0,π] and we need to find the value of f(x) so that it is continuous everywhere in its domain (domain = set of numbers for which f is defined)

As we have expression for x ≠ π/4, which is continuous everywhere in [0,π],so

If we make it continuous at x = π/4 it is continuous everywhere in its domain.

Given,

……….equation 1

Let f(x) is continuous for x = π/4

∴

= = [∵ tan (π/2–θ) = cot θ ]

= [multiplying and dividing by π/4–x and π/2–2x to apply sandwich theorem]

=

[∵

∴ value that can be assigned to f(x) at x = π/4 is