Question

Show that the function defined by is a continuous function.

Answer 1

The expression for the function f is defined as,

f( x )=| cosx |

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 − | cosx | = lim x→c−h | cosx | = lim h→0 | cos( c−h ) | =| cosc | (1)

The right hand limit of the function is,

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

The exact value of the function for x=cis,

f( x=c )=| cosc |(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 of continuity of the function f at x=0 is fulfilled.

Hence, f is a continuous function.