Examine that is a continuous function.
The expression for the function f is defined as,
f ( x ) = sin | x |
Consider a point c within the domain of the function f , to check for continuity of the function.
The left hand limit of the function is,
lim x → c − f ( x ) = lim x → c − sin | x | = lim x → c − h sin | x | = lim h → 0 sin | c − h | = sin | c | (1)
The right hand limit of the function is,
lim x → c + f ( x ) = lim x → c + sin | x | = lim x → c + h sin | x | = lim h → 0 sin | c + h | = sin | c | (2)
The exact value of the function for x = c is,
f ( x = c ) = sin | c | (3)
From equations (1), (2) and (3),
lim x → 0 − f ( x ) = lim x → 0 + f ( x ) = f ( x = 0 ) (4)
From equation (4), the condition for continuity of the function f at x = 0 is fulfilled.
The expression for the function
Consider a point
The left hand limit of the function is,
The right hand limit of the function is,
The exact value of the function for
From equations (1), (2) and (3),
From equation (4), the condition for continuity of the function
