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) = {{:(2x + 1",",  "if"  x ≤ - 1),(3x",",  "if"  - 1 < x < 1),(2x - 1",",  "if"  x ≥ 1):}`

Answer 1

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

= 3 × 1

= 3

`lim_(x -> 1^+) f(x) =  lim_(x -> 1^+) (2x - 1)`

= 2 × 1 – 1

= 2 – 1

= 1 

∴ `lim_(x -> 1^-) f(x) ≠  lim_(x -> 1^+) f(x)`

Hence, `lim_(x -> 1) f(x)` does not exist.

∴ f(x) is not continuous at x = 1.

Let y = `f(x)`

x	– 2	– 1	0	1	2	3
y	2x + 1 – 3	2x + 1 – 1	3x 0	2x – 1 1	2x – 1 3	2x – 1 5