Question

A function f (x) is defined as
f (x) = x + a, x < 0
= x,       0 ≤x ≤ 1
= b- x,   x ≥1
is continuous in its domain.
Find a + b.

Answer 1

f (x) is continuous in its domain.

f (x) is continuous at x = 0 & x = 1
Since f(x) is continuous at x = 0

`therefore lim_(x->0^-)f(x)=lim_(x->0^+)f(x)=f(0)`

`lim_(x->0)(x+a)=lim_(x->0)x=0`

`0+a=0`

`a=0`

Also f (x) is continuous at x = 1

`therefore lim_(x->1^-)f(x)=lim_(x->1^+)f(x)=f(1)`

`lim_(x->1)(x+a)=lim_(x->1)(b-x)=b-1`

`1=b-1`

`b=2`

`a+b=2`