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,
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,
f(x) = \(\frac{tan(\frac{\pi}{4}-x)}{cot\,2x}\) for x ≠ \(\frac{\pi}{4}\) …….equation 1
Let f(x) is continuous for x = π/4
[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 \(\frac{1}{2}\)
