Question

x+1, if x2110,f(x)=

Answer 1

The given function is,

f( x )={ x+1     ,  x≥1 x 2 +1   ,  x<1

Consider k be any real number, then the cases will be k<1 or k=1 or k>1.

When k<1, the function becomes,

f( k )= k 2 +1

The limit of the function is,

lim x→k f( x )= lim x→k ( x 2 +1 ) = k 2 +1

It can be observed that, at k<1, lim x→k f( x )=f( k ).

Therefore, the function is continuous for all points smaller than 1.

When k=1, the left hand limit of the function is,

LHL= lim x→ 1 − f( x ) = lim x→ 1 − ( x 2 +1 ) =1+1 =2

The right hand limit of the function is,

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

It is observed that, LHL=RHL.

Therefore, the function is continuous at k=1.

When k>1, the function becomes,

f( k )=k+1

The limit of the function is,

lim x→k f( x )= lim x→k ( x+1 ) =k+1

It can be observed that, lim x→k f( x )=f( k ).

The function is continuous for all points greater than 1. Therefore, there is no point of discontinuity for the given function.