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प्रश्न
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):}`
उत्तर
|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
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