Question

Show that the function defined by f ( x ) = cos ( x 2 ) is a continuous function.

Answer 1

The expression for the function f is defined as,

f( x )=cos( x 2 )

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 − cos( x 2 ) = lim x→c−h cos( x 2 ) = lim h→0 cos ( c−h ) 2 =cos c 2 (1)

The right hand limit of the function is,

lim x→ c + f( x )= lim x→ c + cos( x 2 ) = lim x→c+h cos( x 2 ) = lim h→0 cos ( c+h ) 2 =cos c 2 (2)

The exact value of the function for x=cis,

f( x=c )=cos( c 2 )(3)

From equation (1),(2) and (3).

lim x→ 0 − f( x )= lim x→ 0 + f( x )=f( x=0 )(4)

From equation (4), the condition of continuity of the function f at x=0 is fulfilled.

Hence, f is a continuous function.