Question

Is the function defined by f(x) = `{(x+5", if"  x <= 1),(x -5", if"  x > 1):}` a continuous function?

Answer 1

f(x) = `{(x+5", if"  x <= 1),(x -5", if"  x > 1):}`

`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (x + 5)

= `lim_(h -> 0)` [1 − h + 5]

= `lim_(h -> 0)` (6 − h)

= 6 − 0

= 6

`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (x − 5)

= `lim_(h -> 0)` (1 + h − 5)

= `lim_(h -> 0)` (h − 4)

= 0 − 4

= −4

Hence, f is not continuous at x = 1.