(i)

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

h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

Function is defined for all real numbers and we need to find the value of k so that it is continuous everywhere in its domain

As, for x ≠ 0 it is just a combination of trigonometric and linear polynomial both of which are continuous everywhere.

As x = 0 is only point at which function is changing its nature so it needs to be continuous here.

f (0) = 3k [using equation 1]

(ii)

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

h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

From equation 1, it is clear that f(x) is changing its expression at x = 2

Given, f (x) is continuous everywhere