Question

Find all points of discontinuity of f, where f is defined by:

f(x) = `{(x/|x|", if"  x<0),(-1", if"  x >= 0):}`

Answer 1

f(x) = `{(x/|x|", if"  x<0),(-1", if"  x >= 0):}`

`lim_(x -> 0^-)` f(x) = `lim_(x -> 0^-) x/abs x`

= `lim_(x -> 0^-) x/(-x)`

= `lim_(x -> 0^-)` (−1)

= −1

`lim_(x -> 0^+)` f(x) = `lim_(x -> 0^+)` (−1) = −1

Also f(0) = −1

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

Hence, f is continuous at x = 0.