Question

Find the point s of discontinuity of f , where

Answer 1

The function is given as,

f(x)={ sinx x ,x<0 x+1,x≥0

Consider k as any real number.

According to the question, k<0 or k=0 or k>0.

In the case when k<0,

f( k )= sink k

Take limits on both sides of the given equation,

lim x→k f( x )= lim x→k ( sinx x ) = sink k

Here,

lim x→k f( x )=f( k )

Hence, the function is continuous for all real numbers less than 0.

In the case when, k=0.

f( 0 )=0+1 =1

Consider the left hand limit,

LHL= lim x→ 0 − f( x ) = lim x→ 0 − ( x+1 ) =0+1 =1

Consider the right hand limit,

RHL= lim x→ 0 + f( x ) = lim x→ 2 + ( x+1 ) =0+1 =1

At x=0, LHL=RHL.

Hence, function is continuous at x=0.

In the case when k>0.

f( k )=k+1

Take limit on both sides of the given equation,

lim x→k f( x )= lim x→k ( x+1 ) =k+1

Here,

lim x→k f( x )=f( k )

Hence, the function is continuous for all real numbers greater than 0.

Thus, the function is continuous for all real numbers and it has no point of discontinuity.