Disucss the continuity of the function f given by
` f (x) = {:{ (-x, if x ge0) , (-x^(2) , if x lt 0):}`

Explanation

The function is defined at every real number. Graph is as shown. The domain of definition of f(x) inot three disjoint subsets of the real line.
Let ` D_(1) = { x in R : x lt 0) `
` D_(2) = {0}`
`D_(3) = { x in R : x gt 0}`
Case I :
x lt 0 , ` f(x) = -x^(2)` the function is continuous at all x lt 0
Case II :
x gt 0 , f(x) = -x the function is continuous at all x gt 0
Case III :
x=0 , f(0)=0
The left hand limit of f(x) at x =0 is
` underset(x to 0^(-))lim f(x) = underset(x to 0^(-)) lim (-x) =0 `
The right hand limit of f(x) at x = 0 is
` underset(x to x^(+)) lim f(x) = lim(x to 0^(+)) ( -x) =0`
The ` underset(x to 0) f(x) = 0= f(0) ` and hence f(x) is continuous at x =0 . This means that f(x) is continuous at every point in its domain. Hence , f(x) is a continuous function.