Question

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

f(x) = `{(|x|/x", if"  x != 0),(0", if"  x = 0):}`

Answer 1

f(x) = `{(|x|/x", if"  x != 0),(0", if"  x = 0):}`

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

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

= `lim_(x -> 0^-)` (−1)

= −1

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

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

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

= 1

Hence, f is not continuous at x = 0.