Show that the function `f(x)={:{(x + lambda, , x lt1),(lamdax^(2)+1, x ge1):}`
is a continuous function , regardles of the regardless of the choice of ` lambda in R `

Explanation

Hint : Note that ` D_(f) = R` Let 'a' be any arbitrary real number.
Case I : Let a lt 1 , then in ` (-oo, 1)`
`lim_(x to a) f(x) =lim_(x to a) (x+ lambda) = a+ lamda=f(a)`
f(x) is continuous at a lt 1
Case II : Let a gt 1 , then in ` (1 , oo)`
`lim_(x toa) f(x) = lim_(x to a) ( lambdax^(2)+1) = lambdaa^(2) +1 = f(a)`
f (x) is contunuous for a gt1
Case III : Let a =1
`lim_(x toa ^(+)) f(x) = lim_(xto1^(+)) (lamdax^(2) +1) = lambda + 1`
` and lim_(x to a^(-)) f(x) = lim_(x to 1^(-)) (x + lambda) = 1 + lambda`
Also , `f(a) = f(1) =lambda(1)^(2) + 1 = lambda+1 `
f(x) is continuous at a =1