A functon f(x) is defined as
`f(x)=x+a, "for" x lt0`
`=x, "for" 0lex lt1,`
`=b-x, "for" x ge 1`
is continous on its domain. Find `a+b.`

Explanation

f is continous on its domain R.
`therefore` it is continous at `x=0` and at `x=1.`
Continuity at `x=0`
`f(x)=x+a, x lt 0`
`thereforeunderset(x to 0)limf(x)=underset(x to 0)lim(x+a)`
`=0+a=a`
`f(x), o le x lt 1`
`therefore f(0) =0`
f is continous at `x=0`
`therefore underset( x to 0^(-))limf(x)=f(0)thereforea=0`
Continuity at `x=1`
From (1), `underset(x to 1^(-))limf(x) =underset(x to 1)limx=1`
`f(x) =b-x, x ge1`
`therefore f(1)=b-1`
f is continuous at `x=1`
`thereforeunderset(x to 1^(-))lim f(x) =f(1)`
`therefore 1=b-1 therefore b=2`
`therefore a+b=0+2=2.`