Question

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= 2x^2 + x + 1",", "for"  |x - 3| ≥ 2),(= x^2 + 3",", "for"  1 < x < 5):}`

Answer 1

|x − 3| ≥ 2

∴ x – 3 ≥ 2 or x – 3 ≤ – 2

∴ x ≥ 5 or x ≤ 1

∴ f(x) `{:(= 2x^2 +  x + 1, ";"  x ≤ 1),(= x^2 + 3, ";"  1 < x < 5),(= 2x^2 + x + 1, ";"  x ≥  5):}`

Consider the intervals

x < 1 i.e. (– ∞, 1)

1 < x < 5 i.e. (1, 5)

x > 5 i.e. (5, ∞)

In all these intervals f(x) is a polynomial function and hence is continuous at all points.

For continuity at x = 1:

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

= 2(1)² + 1 + 1

= 4

`lim_(x -> 1^+) "f"(x) = lim_(x -> 1^+) (x^2 + 3)`

= (1)² + 3

= 4

Also f(1) = 2(1)² + 1 + 1

= 4

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

∴ f(x) is continuous at x = 1

For continuity at x = 5:

`lim_(x -> 5^-) "f"(x) = lim_(x -> 5^-) (x^2 + 3)`

= (5)² + 3

= 28

`lim_(x -> 5^+) "f"(x) = lim_(x -> 5^+) (2x^2 + x + 1)`

= 2(5)² + 5 + 1

= 56

∴ `lim_(x -> 5^-) "f"(x) ≠ lim_(x -> 5+) "f"(x)`

∴ f(x) is discontinuous at x = 5

∴ f(x) is continuous for all x ∈ R, except at x = 5