Question

Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.

`f(x) = {{:((x - 1)^3",",  "if"  x < 0),((x + 1)^3",",  "if"  x ≥ 0):}`

Answer 1

`f(x) = {{:((x - 1)^3",",  "if"  x < 0),((x + 1)^3",",  "if"  x ≥ 0):}`

`lim_(x -> 0^-) f(x) =  lim_(x -> 0^-) (x - 1)^3`

= (0 – 1)³

= – 1  .........(1)

`lim_(x -> 0^+) f(x) =  lim_(x -> 0^+) (x + 1)^3`

= (0 + 1)³

= 1  .........(2)

From equation (1) and (2) we have

`lim_(x -> 0^-) f(x) ≠  lim_(x -> 0^+) f(x)`

∴ `lim_(x -> 0) f(x)` does not exist.

Hence f(x) is not continuous at x = 0.

x	– 2	– 2	0	1	2
y	(x – 1)3 – 8	(x – 1)3 – 27	(x + 1)3 1	(x + 1)3 8	(x + 1)3 27